UNIVERSITEIT GENT
GHENT UNIVERSITY
FACULTEIT ECONOMIE EN BEDRIJFSKUNDE
FACULTY OF ECONOMICS AND BUSINESS
ADMINISTRATION
ACADEMIC YEAR
2015 – 2016
OPTIMAL CUSTOMER BEHAVIOUR IN
OBSERVABLE AND UNOBSERVABLE
QUEUEING SYSTEMS
Master’s Dissertation submitted to obtain the degree of
Master of Science in Business Engineering
Pieter Verbeke
Under the guidance of
Prof. Dr. Ir. Herwig Bruneel and Dr. Bart Steyaert
UNIVERSITEIT GENT
GHENT UNIVERSITY
FACULTEIT ECONOMIE EN BEDRIJFSKUNDE
FACULTY OF ECONOMICS AND BUSINESS
ADMINISTRATION
ACADEMIC YEAR
2015 – 2016
OPTIMAL CUSTOMER BEHAVIOUR IN
OBSERVABLE AND UNOBSERVABLE
QUEUEING SYSTEMS
Master’s Dissertation submitted to obtain the degree of
Master of Science in Business Engineering
Pieter Verbeke
Under the guidance of
Prof. Dr. Ir. Herwig Bruneel and Dr. Bart Steyaert
Permission
I, the undersigned, declare that the content of this Master’s thesis can be consulted and/or
reproduced, provided chapter and verse are clearly referenced.
Pieter Verbeke
I
II
Abstract
There are numerous situations in which customers have to queue before they can receive a
certain service. Customers can initially make the decision to either join or not join the queue
(joining vs. balking). In cases where the queue can be observed, this decision depends mostly
on the length of this queue, compared to the customer’s perceived value of the service. In some
situations, however, the customer does not have access to this information, which leads to a
different strategy for decision making.
We have found that, for several systems, individual customer behaviour differs from what
is socially desired. This problem can be solved by imposing appropriate admission fees or
waiting tolls. When comparing observable and unobservable systems, results on whether or
not to reveal the queue length are not unambiguous. All depends on the specific values of the
system parameters. Additional variation in the arrival or service process leads to lower optimal
queue-joining rates and therefore loss of revenue for the service provider.
Keywords: joining, balking, individual behaviour, social behaviour, observable queue, unobservable queue, M M 1, M M s, H2 M 1, M H2 1.
III
IV
Preface
After two years of hard work and countless sheets of draft paper, I can proudly present to the
reader the Magnum Opus of my academic career. Evidently, I would not have been able to
complete this work without the help of all the people that are close to my heart. To them I
wish to express my sincere word of thanks.
Foremost, I would like to thank my parents for providing me with support throughout the
course of this work and my life so far. They have always made sure that I have a safe place to
fall back to when times are rough. And maybe even more importantly, have kept my feet on
the ground when I started to drift away.
A second word of thanks goes out to my friends for providing some much needed distraction. Together we have experienced awesome times, which gave me the energy needed to climb
the mountain that this work has proved to be. And to those friends who happen to be my
colleagues and are also writing their thesis... Thank you for bringing in the right amount of
stress and pressure to get me started.
Finally, I would like to thank everyone that was involved in the making of this work. A
specific mention goes out to Bart Steyaert, who has answered even my smallest questions and
has accommodated me with feedback throughout the course of creating this work.
And to the reader, whoever you may be, I can only say: Enjoy!
May 17, 2016, Ghent
Pieter Verbeke
V
VI
Contents
1 Introduction
1.1 Problem Definition and Research Questions . . . . . . . . . . . . . . . . . . . .
1.2 Structure of the Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2 Single-Server Systems with Exponentially
vice Times (M/M/1)
2.1 Introduction . . . . . . . . . . . . . . . . .
2.2 Observable System . . . . . . . . . . . . .
2.2.1 Individual Behaviour . . . . . . . .
2.2.2 Social Behaviour . . . . . . . . . .
2.2.3 Profit Maximisation . . . . . . . .
2.3 Unobservable System . . . . . . . . . . . .
2.3.1 Individual Behaviour . . . . . . . .
2.3.2 Social Behaviour . . . . . . . . . .
2.3.3 Profit Maximisation . . . . . . . .
2.4 Observable vs. Unobservable Systems . . .
2.4.1 Individual Behaviour . . . . . . . .
2.4.2 Social Behaviour . . . . . . . . . .
2.4.3 Profit Maximisation . . . . . . . .
Distributed Interarrival and Ser-
3 Multi-Server Systems with Exponentially
vice Times (M/M/s)
3.1 Introduction . . . . . . . . . . . . . . . . .
3.2 Observable System . . . . . . . . . . . . .
3.2.1 Individual Behaviour . . . . . . . .
3.2.2 Social Behaviour . . . . . . . . . .
3.2.3 Profit Maximisation . . . . . . . .
3.3 Unobservable System . . . . . . . . . . . .
3.3.1 Individual Behaviour . . . . . . . .
3.3.2 Social Behaviour . . . . . . . . . .
3.3.3 Profit Maximisation . . . . . . . .
Distributed Interarrival and Ser-
VII
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1
1
3
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27
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30
3.4
3.5
Observable vs. Unobservable Systems
3.4.1 Individual Behaviour . . . . .
3.4.2 Social Behaviour . . . . . . .
3.4.3 Profit Maximisation . . . . .
Single-Server vs Multi-Server Systems
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31
31
32
33
34
4 Single-Server Systems with Hyperexponentially Distributed Interarrival Times
(H2 /M/1)
36
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
4.2 Observable System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
4.2.1 Individual Behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.2.2 Social Behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.2.3 Profit Maximisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.3 Unobservable System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.3.1 Individual Behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.3.2 Social Behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.3.3 Profit Maximisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.4 Observable vs. Unobservable Systems . . . . . . . . . . . . . . . . . . . . . . . . 49
4.4.1 Individual Behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.4.2 Social Behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.4.3 Profit Maximisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.5 Effect of Additional Variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
5 Single-Server Systems with Hyperexponentially
(M/H2 /1)
5.1 Introduction . . . . . . . . . . . . . . . . . . . . .
5.2 Observable System . . . . . . . . . . . . . . . . .
5.2.1 Individual Behaviour . . . . . . . . . . . .
5.2.2 Social Behaviour . . . . . . . . . . . . . .
5.2.3 Profit Maximisation . . . . . . . . . . . .
5.3 Unobservable System . . . . . . . . . . . . . . . .
5.3.1 Individual Behaviour . . . . . . . . . . . .
5.3.2 Social Behaviour . . . . . . . . . . . . . .
5.3.3 Profit Maximisation . . . . . . . . . . . .
5.4 Observable vs. Unobservable Systems . . . . . . .
5.4.1 Individual Behaviour . . . . . . . . . . . .
5.4.2 Social Behaviour . . . . . . . . . . . . . .
5.4.3 Profit Maximisation . . . . . . . . . . . .
5.5 Effect of Additional Variation . . . . . . . . . . .
VIII
Distributed Service Times
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55
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6 Numerical Procedures
72
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
6.1.1 Social Optimisation in an Observable System . . . . . . . . . . . . . . . 72
6.1.2 Individual Optimisation in an Unobservable System . . . . . . . . . . . . 73
6.1.3 Social Optimisation in an Unobservable System . . . . . . . . . . . . . . 73
6.1.4 Numerical Solution for a Set of Balance Equations . . . . . . . . . . . . . 74
6.2 Single-Server Systems with Exponentially Distributed Interarrival and Service
Times (M/M/1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
6.2.1 Social Optimisation in an Observable M/M/1/n-system . . . . . . . . . 75
6.3 Multi-Server Systems with Exponentially Distributed Interarrival and Service
Times (M/M/s) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
6.3.1 Social Optimisation in an Observable M/M/s/n-system . . . . . . . . . 75
6.3.2 Individual Optimisation in an Unobservable M/M/s-system . . . . . . . 76
6.3.3 Social Optimisation in an Unobservable M/M/s-system . . . . . . . . . . 77
6.4 Single-Server Systems with Hyperexponentially Distributed Interarrival Times
(H2 /M/1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
6.4.1 Numerical Solution for H2 /M/1/n . . . . . . . . . . . . . . . . . . . . . . 78
6.4.2 Individual Optimisation in an Unobservable H2 /M/1-system . . . . . . . 79
6.4.3 Social Optimisation in an Unobservable H2 /M/1-system . . . . . . . . . 80
6.5 Single-Server Systems with Hyperexponentially Distributed Service Times (M/H2 /1) 81
6.5.1 Numerical Solution for M/H2 /1/n . . . . . . . . . . . . . . . . . . . . . 81
6.5.2 Individual Optimisation in an Unobservable M/H2 /1-system . . . . . . . 82
6.5.3 Social Optimisation in an Unobservable M/H2 /1-system . . . . . . . . . 83
7 Conclusion
7.1 Individual vs. Social Behaviour . . .
7.2 Observable vs. Unobservable Systems
7.3 Effect of Additional Variation . . . .
7.4 Further Research . . . . . . . . . . .
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IX
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X
Chapter 1
Introduction
Before one can start delving into this thesis, it is important to know its scope and relevance
in the academic literature. Therefore the problem is to be unambiguously defined in this
introductory chapter, as well as some specific research questions that make clear what answers
can and cannot be expected to be found in this work.
Furthermore, a general overview of the analysis is given, which structures and incorporates
existing literature. This is the basis upon which all other chapters are built. To make life easier
for the reader, all notations used throughout this work are bundled in this chapter.
1.1
Problem Definition and Research Questions
There are numerous situations in which customers have to queue before they can receive a
certain service. Customers can initially make the decision to either join or not join the queue
(joining vs. balking). In cases where the queue can be observed, this decision depends mostly
on the length of this queue, compared to the customer’s perceived value of the service. In some
situations, however, the customer does not have access to this information, which leads to a
different strategy for decision making. The goal of this thesis is to derive, for several systems,
the optimal strategy to follow and how to maximise profits as a service provider, given these
strategies.
Surely, it is not hard to comprehend that customers will not always take the socially desired
actions. It is said that customer joining rates are individually optimal if they occur under Nash
Equilibrium conditions, meaning that no customer has an incentive to change their strategy
given the other customer’s strategies [10]. A customer wants to minimise his own time in queue
and be served as quickly as possible. In order to optimise total welfare, the queueing time of all
customers needs to be minimised. This is, in truth, the service provider’s point of view, since
he wants to maximise the throughput in his system.
1
We can therefore distinguish two types of behaviour, which we will call individual and social
behaviour respectively. When both types lead to the same decision by an individual customer,
no interference is needed. If they differ, meaning that the individually optimal behaviour is
suboptimal from a social point of view, actions will have to be taken to tempt the customer
into taking a socially optimal decision.
Our first research question can thus be formulated as follows:
”For several queueing systems, is there a difference between Social and Individual Optimality?
If so, what actions can be taken to make these optima compatible and what is the impact on
system parameters?”
Other factors regarding the decision making process also have to be taken into account. In
particular, the focus of this thesis lies on the differences between systems where the queue can
be observed and systems where this is not the case. For example, waiting at a bakery shop is an
observable queueing system, whereas being placed ’on-hold’ in a call-center is an unobservable
queue. In the latter, less information is available for the customer, which, in some situations,
leads to suboptimal decision-making. Since we are aiming for optimality, it might be beneficial
to make invisible queues observable for the customer (e.g. indicating how many customers are
already in queue).
This leads us to stipulate a second research question:
”For several queueing systems, is it worthwhile to make information on the queue length
available to customers in order to accomplish optimality? If so, what is the impact on system
parameters”
Another interesting aspect is the effect of variation on these decisions. Most common, the
interarrival and service times are modeled using exponential distributions that are typified by
a single parameter. We would like to know what happens in a situation with the same average
arrival or service rate, but with higher variation. This induces us to formulate a third and final
research question:
For several queueing systems, what is the effect of additional variation on the system
parameters? How does this affect the tactic employed by the service provider?
In order to answer these three questions, it is necessary to have a common understanding of
several key concepts that are specific to this field of study. It is assumed that the reader
has a notion of basic queueing theory terminology, which will not be explained specifically.
Some more general concepts, along with the structure of our analysis, are to be clarified in the
following section.
2
1.2
Structure of the Analysis
As is made clear in the problem definition above, we can structure the analysis on two dimensions: the type of system under consideration (observable vs. unobservable) and the behaviour
of customers (individual vs. social) [5]. Figure 1.1 graphically depicts this, including the
notation for the effective queue-joining rates that will be used throughout this paper.
Individual Behaviour
O,I
λU,I
e
λe
Observable System
Unobservable System
O,S
λU,S
e
λe
Social Behaviour
Figure 1.1 Notation for Effective Queue-joining rates
In order to be able to compare these rates, it is important to clearly define several parameters
that have the same meaning across all systems and for both types of behaviour.
Table 1.1 gives a summary of these generally used parameters. Note that it is actually not
necessary in the expression of ρ to make a distinction between the single- and multi-server case
(for s = 1, both formulas are equal), but for the sake of clarity, we have split these up.
λ
µ
R
C
G
s
ρ
rate at which customers arrive in the system
rate at which customers are served
gain for a completed service
cost of waiting per time unit
net gains for waiting for the service
number of servers in the service unit
utilisation rate of the system with infinite storage capacity
= µλ , for single-server systems
=
λe
ρe
λ
,
sµ
for multi-server systems
effective queue-joining rate, rate at which customers leave the service unit
utilisation rate of the service unit, taking into account the effect of balking
= λµe , for single-server systems
=
λe
,
sµ
for multi-server systems
Table 1.1 General System Parameters
The parameters R and C are specific to the problem we are considering. Throughout this work
we are assuming that Rµ
≥ 1. Otherwise, a customer would not even want to wait for his own
C
service, which leads to trivial solutions [3]. Another assumption is that the utility functions of
individual customers are identical an additive, i.e. there is no difference in preferences among
3
customers. In addition, reneging is not allowed. Once a customer joins he has to wait for the
service.
A customer’s decision to join or balk will always depend on the perceived benefit of completing the service R and the cost of waiting C, expressed per time unit. Joining will only
happen if he perceives his net gains G to be positive. It is with respect to this problem that a
distinction between the four cases should be made.
Observable System
In an observable system, the customer has information on the queue length when he makes his
decision to join or not. There is a cut-off value at which he perceives his net gains no longer
to be positive and at which he decides to balk. It is said that the customers each follow a pure
threshold strategy [3]. This entails that their actions, no matter what the state of the queue
may be, can be determined up front since there is only one optimal decision to make. The
strategy is therefore not a random choice. This reasoning is of course only true under a FCFS
queueing discipline [3]. Table 1.2 shows the parameters that are used in the analysis of this
type of system.
n
ni
ns
Gi (n)
Gs
qn
Ln
number of customers in queue upon arrival, including the arriving customer
optimal threshold value under individual behaviour
optimal threshold value under social behaviour
net gains per customer under individual behaviour (function of n)
net gains per time unit under social behaviour
loss probability, probability of a full buffer of size n upon arrival
expected number of customers in the system
Table 1.2 Parameters used in the analysis of Observable Systems
Under individual behaviour, in a single-server environment [9], a customer’s net gains are given
by
C
Gi (n) = R − n .
µ
The largest n for which Gi (n) ≥ 0, is easily determined by solving the inequality and keeping
in mind that n takes an integer value [9]. Therefore it is clear that ni is given by
ni = b
Rµ
c.
C
(1.1)
Under this strategy, an arriving customer balks if he observes ni or more customers in queue
and only joins if there are ni − 1 or less [9]. The number of customers in queue will therefore
never exceed ni . In observable systems, our analysis will hence always be based on a system
with a finite queue!
4
In the case of multiple servers [6], the expression for the net gains can be stated as
Gi (n) = R −
(n − s)+
+1
s
C
,
µ
where (n − s)+ denotes max{0, n − s}. For cases in which n < s, the customer does not have
to queue and can immediately enter the service unit. Otherwise, the customer has to wait
1
according to his position in queue. Since there are multiple servers, his waiting time will be sµ
per customer in front of him, leading to the above equation.
The optimal threshold value ni is thus given by
ni = b
Rsµ
c.
C
(1.2)
Remark that for s = 1 the same result as in equation (1.1) is reached. Also note that these
expressions do not feature the arrival intensity λ. It is therefore highly doubtful that this strategy is in unison with the socially optimal one [3].
Under social behaviour, the net gains for all customers are maximised. We therefore need
to aggregate all benefits and all costs to determine Gs [3]. The benefits are determined by
multiplying the effective joining rate with the gain of the service. The total cost per time unit
depends on the number of customers in the system. Therefore, the net social gains per time
unit are given by
Gs = λR(1 − qn ) − CLn .
(1.3)
Trying to mathematically find the optimal n for a specific system will prove to be very hard or
even impossible. We will therefore need to resort to numerical procedures in order to determine
ns . The precise methods hereto are explained in Chapter 6.
Knowing these optimal threshold values, it is rather easy to compute the resulting effective
queue-joining rates by multiplying the system arrival rate λ with the loss probability under the
given threshold [5].

λO,I = λ(1 − q )
ni
e
λO,S = λ(1 − q ).
ns
e
Unobservable System
The strategy followed in unobservable systems is completely different in nature. Customers now
cannot incorporate information on the queue length in their decision to join or balk. Therefore,
they will all adopt the same randomized strategy, specified by a probability p ∈ [0, 1], where
5
p indicates the probability of joining and (1 − p) the probability of balking [2]. The effective
queue-joining rate λUe is thus equal to pλ. Notice that the queueing discipline is of no importance
here! The restriction of FCFS can thus be left out in an unobservable system [3]. Similar to
the previous paragraph, Table 1.3 gives an overview of the parameters that are used.
p
Gi
Gs
Ts (λ)
λ∗I
λ∗S
probability of joining the queue
net gains per customer under individual behaviour
net gains per time unit under social behaviour
expected time in the system (function of λ)
unrestricted optimal effective queue-joining rate under individual behaviour
unrestricted optimal effective queue-joining rate under social behaviour
Table 1.3 Parameters used in the analysis of Unobservable Systems
There are three possible strategies to follow: always balk (p = 0), always join (p = 1) or a mixed
strategy (0 < p < 1) [3]. If every customer balks, it is clear that λUe = 0. In the other extreme
case, i.e. always joining, λUe = λ. It is the mixed strategy that requires a little more analysis [2].
A customer’s individual net gains Gi can be stated as
Gi = R − C.Ts (λ),
(1.4)
Note that in the unobservable case there is no restriction on the number of customers in queue.
We therefore use the results for Ts (λ) from systems with infinite queueing space.
In order to find the optimal queue-joining rate, it must be that Gi ≥ 0. Solving for a specific system will then lead to an expression for the optimal λ∗I . In some systems, however, we
will not be able to compute this mathematically, inducing us to use numerical procedures.
The optimal queue-joining rate combines all three strategies [2] and can be formulated as
= min{λ, λ∗I }.
λU,I
e
(1.5)
The same three strategies are considered under social behaviour, but now we need to maximise
the total throughput to find the optimal rate [3]. In parameters, the total net gains Gs are
determined by
Gs = λ[R − C.Ts (λ)].
(1.6)
After optimising the social gains we will end up with an expression (or a result from the
numerical procedure) for λ∗S [2]. Similar to the individual case, we can state the combination
of all three strategies as
λU,S
= min{λ, λ∗S }.
(1.7)
e
6
Results
Once we have the four effective-queue joining rates that represent the four cases of Figure 1.1,
we can start comparing and draw conclusions [5].
A first comparison entails individual versus social behaviour, which is nothing more than a
profit maximisation problem. The service provider needs to find a way to make individual customers act in a socially optimal way. This can either be achieved by decreasing R or increasing
C. In practice this can be done by respectively imposing an admission fee a or a waiting toll t
[3]. These will result in the equality of individual and social effective-queue joining rates.
The second dimension in which to compare is the type of system under consideration. In
other words, this allows us to determine whether or not it is beneficial to reveal information on
the queue length to customers.
Chapters 4 and 5 deal with the third research question. By respectively modeling the
interarrival and service times using a hyperexponential distribution, the effect of additional
variation in the arrival and service process can be determined.
7
Chapter 2
Single-Server Systems with
Exponentially Distributed Interarrival
and Service Times (M/M/1)
2.1
Introduction
Now let’s get started! Commencing with the most stringent and therefore easiest to solve system, certain assumptions will be gradually relaxed or left out to come to a more general system.
More precisely, we start with the so-called M/M/1-system, followed by M/M/s, H2 /M/1 and
M/H2 /1. It is not intended to derive each model in detail, the focus lies on the interpretation
of results and the conclusions that can be drawn from them.
2.2
Observable System
This is the base system developed by P. Naor [9] that is the foundation for all further research
in the field of behavioural queueing theory. To unambiguously define the system under consideration, we will consistently list a number of assumptions, following R. Hassin and M. Haviv
[3]. This observable M/M/1-system is defined by the following 10 assumptions.
1. Poisson arrival stream of customers with rate λ, arriving at a single server.
2. Service times are i.i.d. exponentially distributed with rate µ.
3. The benefit for a customer from completed service is R.
4. The cost for a customer to stay in the system is C per unit of time.
5. Customers are risk neutral, meaning they maximise the expected value of their net benefit.
6. Utility functions of individual customers are identical and additive, i.e. there are no
differences in preference among customers.
8
7. R ≥ Cµ , in order to avoid the case where a customer would be unwilling to wait even for
his own service. This would lead to trivial solutions.
8. The service discipline is FCFS.
9. Reneging is not allowed, once a customer joins the queue he has to wait for the service.
10. The only decision a customer has to make is whether to join or balk. Once balked, the
customer cannot return to the queue.
2.2.1
Individual Behaviour
Expression (1.1) provides us with the optimal threshold value under individual behaviour. The
only parameter that is to be determined, is the loss probability qn in an M/M/1/n-system.
Remember that the threshold strategy in an observable system implies that there is only a
limited number of customers allowed in queue. Markovian analysis of this system [1] has shown
that

 (1−ρ)ρn
, ρ 6= 1
n+1
qn = 1−ρ
 1
, ρ = 1.
n+1
The effective queue-joining rates are determined by multiplying the system arrival rate λ with
the probability of joining (i.e. 1 − qn ), resulting in
λO,I
= λ(
e
1 − ρni
)
1 − ρni +1
, ρ 6= 1,
(2.1)
λO,I
= λ(
e
ni
)
ni + 1
, ρ = 1,
(2.2)
and
with ni = b Rµ
c.
C
2.2.2
Social Behaviour
In order to come to results under socially optimal behaviour, we only need an expression for
the expected number of customers in the system Ln . Again, this can be found in literature on
traditional queueing analysis [1]. It is shown that
Ln =


ρ
1−ρ
−
(n+1)ρn+1
1−ρn+1
n
, ρ 6= 1
, ρ = 1.
2
9
Making use of expression (1.3) for the net social gains, this means that the socially optimal
strategy ns is the maximiser of
Gs = λR
(n + 1)ρn+1
1 − ρn
ρ
−
−
C[
]
1 − ρn+1
1−ρ
1 − ρn+1
, ρ 6= 1.
Gs is maximised by setting ns equal to bν0 c, where ν0 satisfies [5]:
Rµ
ν0 (1 − ρ) − ρ(1 − ρν0 )
=
(1 − ρ)2
C
, ρ 6= 1.
For ρ = 1 this expression reduces to
ν0 (ν0 + 1)
Rµ
=
2
C
, ρ = 1.
The equation for Gs is not easily solved, but it can be proved mathematically [9] that this expression is unimodal in n, and that the optimal ns ≤ ni . This statement is checked in Section
6.2.1 using numerical procedures.
Figure 2.1 shows the evolution of Gs in function of n, using the outcomes of the numerical procedure. We choose to set R = C = 10, λ = 8 and µ = 10. It is apparent that this
function indeed reaches one and only one maximum value. This means that the threshold value
ns is unambiguously defined.
60
Net Social Gains
50
40
30
20
10
0
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
Threshold value n
Figure 2.1 Evolution of net social gains in an Observable M/M/1-system, R = C = 10, λ = 8, µ = 10
10
The optimal threshold values are plotted in Figure 2.2 as a function of λ. The statement that
ns ≤ ni clearly holds. In fact, the gap between the two only gets bigger for higher utilisation
rates.
12
Threshold value n
10
8
6
Individual Threshold
Social Threshold
4
2
0
0
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18 19 20
λ
Figure 2.2 Optimal Threshold values in an M/M/1-system,
R
C
= 1, µ = 10
Similar to the previous section, the effective queue-joining rates are defined as follows
λO,S
= λ(
e
1 − ρns
)
1 − ρns +1
, ρ 6= 1,
(2.3)
λO,S
= λ(
e
ns
)
ns + 1
, ρ = 1.
(2.4)
and
These can now be used to compare with equations (2.1) and (2.2) to find whether individual
behaviour leads to a socially suboptimal rate.
Using the numerical data, Figure 2.3 shows this difference between individual and social
behaviour while letting the system arrival rate λ (and therefore ρ) vary, whereas in Figure 2.4
the relative benefit of the service compared to its cost fluctuates. It is clear to see that indeed
λO,I
≥ λO,S
e
e , meaning that selfish behaviour leads to higher arrival rates and thus longer queues
and queueing times.
11
For very high values of λ (indicating a very high system utilisation rate ρ), both rates will
converge. The resulting effective queue-joining rate will then be equal to 10, which is the
consequence of a server utilisation rate ρe of 1.
Effective Queue-joining rates
12
10
8
6
Individual Rate
4
Social Rate
2
0
0
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19 20
λ
Figure 2.3 Individual vs. Social Optimisation in an Observable M/M/1-system,
R
C
= 1, µ = 10
If the relative benefit of the service is so high that no customer wants to balk (λe = λ), both
rates converge. For our choice of parameters, this means that the effective queue-joining rate
will be 8.
Effective Queue-joining rates
9
8
7
6
Individual Rate
Social Rate
5
4
3
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
Rμ/C
Figure 2.4 Individual vs. Social Optimisation in an Observable M/M/1-system, ρ = 0.8, µ = 10
12
2.2.3
Profit Maximisation
The fact that selfishly acting customers decrease total welfare comes as no surprise. The question arises as to how to make customers act in a socially optimal way to maximise the revenue of
the service provider. Expressed in parameters, ni needs to decrease in order for ns = ni . There
are two ways to come to this optimum [3], either by decreasing R or increasing C. These payments do not increase social welfare and therefore need not to be included in the profit function.
A first option is to impose an admission fee a, chosen in such a way that
ns = b
(R − a)µ
c.
C
A second possibility is to impose a toll t on waiting. In this case the optimal threshold value is
ns = b
Rµ
c.
C +t
The size of these payments depends on the specific values of R and C, and not solely on their
ratio (as was the case above). In order to demonstrate this graphically, C is held constant and
R is chosen to be variable. The result is depicted in Figure 2.5.
25
Size of fee / toll
20
15
Admission Fee
10
Waiting Toll
5
0
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19 20
Benefit of the Service R
Figure 2.5 Size of the Admission Fee/Waiting Toll in an Observable M/M/1-system, C = 10, µ = 10
There are two reasons to prefer admission fees to waiting tolls. Firstly, the size of a profitmaximising toll is higher than an equivalent admission fee. Although these payments do not
alter social welfare, it is obvious that lower transfers are desired. Secondly, it is apparent that
waiting tolls are much more volatile compared to admission fees. This is especially the case
for systems that have a low (relative) benefit. Since it is hard to estimate the exact costs and
benefits of a specific system, it is advised to opt for an admission fee rather than a waiting toll.
An incorrect estimation of the system parameters will in this case not necessarily lead to a big
error.
13
2.3
Unobservable System
In the unobservable case, assumptions 8 and 10 (Section 2.2) are altered [2]. More precisely,
8. There is no restriction on the service discipline.
10. The customer has a predefined chance to join the queue. Once balked, the customer cannot return to the queue.
As mentioned in the introduction, these changes have a huge impact on the calculations for
determining optimal behaviour.
The only additional parameter needed in the upcoming analysis is the expected time in the
system Ts (λ). Since we are dealing with a straightforward M/M/1-system [1], we know that
Ts (λ) =
2.3.1
1
.
µ−λ
Individual Behaviour
Making use of expression (1.4) and substituting Ts (λ) with the equation above, we immediately
get to a solution for λ∗I [5]. More precisely,
λ∗I = µ −
C
.
R
The optimal effective queue-joining rate is thus given by
λU,I
= min{µ −
e
2.3.2
C
, λ}.
R
(2.5)
Social Behaviour
Determining the socially optimal behaviour is in no sense harder than in the individual case
[5]. Expression (1.6) shows us that λ∗S will be
s
λ∗S = µ(1 −
C
).
Rµ
Again, we can formally state the effective queue-joining rate as
s
λU,S
= min{µ(1 −
e
C
), λ}.
Rµ
(2.6)
It is apparent that λU,I
≥ λU,S
e
e , given the assumptions about R, C and µ. All else being equal,
individual optimisation thus leads to longer queues and longer queueing times than in the so14
cially desired optimum. This is in alignment with the results found in the observable case in
the previous section.
Figures 2.6 and 2.7 illustrate this graphically, keeping either the relative benefit of the
service CR or the utilization factor ρ constant. The service rate µ is never changed since we only
focus on customer-related variables.
10
Effective Queue-joining rates
9
8
7
6
5
Individual Rate
4
Social Rate
3
2
1
0
0
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19 20
λ
Figure 2.6 Individual vs. Social Optimisation in an Unobservable M/M/1-system,
R
C
= 1, µ = 10
In the social optimum, there is no balking until the system utilisation rate ρ reaches 0.68. From
that point on, customers will always balk. The equivalent individual balking point is at higher
utilisation rate, namely 0.9. Both rates will therefore never converge once they have passed the
social optimum.
Effective Queue-joining rates
9
8
7
6
5
Individual Rate
4
Social Rate
3
2
1
0
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
Rμ/C
Figure 2.7 Individual vs. Social Optimisation in an Unobservable M/M/1-system, ρ = 0.8, µ = 10
If the relative benefit of the service is so high that no customer wants to balk (λe = λ), both
rates converge. For our choice of parameters, this means the effective joining rate will be 8.
15
2.3.3
Profit Maximisation
Keeping in mind this gap between the individual and social strategy, the next logical step is to
determine the possibilities to make individual customers act in the socially optimal way with
the purpose of maximising the income for the service provider [2]. Again, we consider two
options to reduce the individual effective queue-joining rate λU,I
e : either asking an admission
fee a or a toll on waiting t. These payments set the costs equal to the benefits of the service.
As was the case with the observable case, they do not add to social welfare (since it are mere
transfer payments) and thus need not to be included in the profit function.
For the admission fee a, under the socially optimal rate, this is
a + C.Ts (λ∗S ) = R.
Therefore, using the expression we derived for λ∗S
s
a=R−
CR
.
µ
Analogously, for the waiting toll t, under the socially optimal rate, this is
(C + t).Ts (λ∗S ) = R.
Filling in the expression for λ∗S results in
t=
p
CRµ − C.
The size of these payments depends on the specific values of R and C, and not solely on their
ratio (as was the case above). In order to demonstrate this graphically, C is held constant and
R is chosen to be variable.
The result is depicted in Figure 2.8. Similar to our findings in Section 2.2.3, admission
fees are lower than comparable waiting tolls. Due to the continuous behaviour of the effective
queue-joining rates, we do not observe the volatile conduct of the waiting tolls.
16
40
35
Size of fee / toll
30
25
20
Admission Fee
15
Waiting Toll
10
5
0
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19 20
Benefit of the Service R
Figure 2.8 Size of the Admission Fee/Waiting Toll in an Unobservable M/M/1-system, C = 10, µ = 10
2.4
Observable vs. Unobservable Systems
Now that optimality in both observable and unobservable queueing systems is determined, the
results for each system can be compared, using equations (2.1) to (2.6). We are now able
to answer the third research question: Is it worthwhile to make an invisible queue visible to
customers in order to accomplish optimality, and what is the impact on queueing parameters?
2.4.1
Individual Behaviour
Similar to what has been done before, Figures 2.9 and 2.10 describe the comparison between
an observable and an unobservable M/M/1-system. What is different here, is that one system
is not indisputably better than the other. All depends on the specific values that the system
parameters take.
At first glance, it seems that there are four possible combinations of equations (2.1), (2.2)
and (2.5) at which the effective rates can be equal (λO,I
= λU,I
e
e ). However, there is only one
option that yields a feasible solution [5], namely that one for which
C
1 − ρni
)
=
µ
−
λ(
1 − ρni +1
R
This equation can only hold if µ −
C
R
, ρ 6= 1.
≤ λ, therefore
1 − ρni
≤1
1 − ρni +1
17
, ρ 6= 1.
Keeping in mind that utilisation rates are non-zero and strictly positive, it is clear that
0 < ρi < 1.
Using this result in the expressions for the effective arrival rates, a necessary and sufficient
can be formulated as follows:
and λU,I
condition for equality of λO,I
e
e
1 − ρni i +1
Rµ
=
.
1 − ρi
C
Effective Queue-joining rates
12
10
8
6
Observable Rate
4
Unobservable Rate
2
0
0
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18 19 20
λ
Figure 2.9 Individual Optimisation in Observable vs. Unobservable M/M/1-systems,
R
C
= 1, µ = 10
Figure 2.9 clearly demonstrates the boundaries for ρ that we just derived. For this choice of
parameters, the intersection is at a utilisation rate of ρ = 0.98. For higher values, both the
observable and unobservable rate converge to their asymptotic values for λIe , which are 10 and
9 respectively.
Effective Queue-joining rates
9
8
7
6
5
4
Observable Rate
3
Unobservable Rate
2
1
0
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
Rμ/C
Figure 2.10 Individual Optimisation in Observable vs. Unobservable M/M/1-systems, ρ = 0.8, µ = 10
18
As is made clear by Figure 2.10 this value for Rµ
, given a certain ρ, need not to be unique,
C
is
hence several intersection points can be distinguished. On the other hand, if a value for Rµ
C
set in advance, it is mere calculus to determine the single resulting utilisation rate. For very
high service gains, both rates converge. This is because customers will always decide to join,
regardless of the fact whether they have information on the queue length or not. For our choice
of parameters, the resulting effective queue-joining rate is 8.
2.4.2
Social Behaviour
A socially optimal strategy does also not readily provide us with a clear answer to what type of
system is best. Again, we turn to previously found equations. Formulas (2.3), (2.4) and (2.6)
= λU,S
are used to find the intersections between the observable and unobservable rates (λO,S
e ).
e
Only one of four possibilities results in a feasible solution:
ns
λ(
1−ρ
) = µ(1 −
1 − ρns +1
This equation can only hold if µ(1 −
q
C
)
Rµ
s
C
)
Rµ
, ρ 6= 1.
≤ λ, therefore
1 − ρns
≤1
1 − ρns +1
, ρ 6= 1.
Keeping in mind that utilisation rates are non-zero and strictly positive, it is clear that
0 < ρs < 1.
Clearly there is a certain analogy with the previous section! This is because the value of n is
not explicitly used. Applying this restriction in the expressions for the effective arrival rates,
we can formally state a necessary and sufficient condition for equality of λO,S
and λU,S
e
e :
1 − ρns s +1
=
1 − ρs
r
Rµ
.
C
The graphical representation of all this is depicted in Figures 2.11 and 2.12, keeping either
R
or ρ constant.
C
19
10
Effective Queue-joining rates
9
8
7
6
5
Observable Rate
4
Unobservable Rate
3
2
1
0
0
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18 19 20
λ
Figure 2.11 Social Optimisation in Observable vs. Unobservable M/M/1-systems,
R
C
= 1, µ = 10
The intersection of the observable and unobservable rate in Figure 2.11 lies at a utilisation
rate ρ = 0.77. This is significantly lower than in the individual case, as we expected from the
mathematical reasoning above.
Effective Queue-joining rates
9
8
7
6
5
Observable Rate
4
Unobservable Rate
3
2
1
0
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
Rμ/C
Figure 2.12 Social Optimisation in Observable vs. Unobservable M/M/1-systems, ρ = 0.8, µ = 10
For very high service gains, both rates converge. This is because customers will always decide
to join, regardless of the fact whether they have information on the queue length or not. For
our choice of parameters, the resulting effective queue-joining rate is 8.
20
2.4.3
Profit Maximisation
It is up to the service provider to decide whether or not to transform his unobservable system
into an observable one by making information on the queue length available to customers.
As has been pointed out before, there is no generally optimal strategy to follow. The exact
revenue-maximising decision depends on the particular values of the system’s parameters. More
precisely, the service provider will try to increase the queue-joining rate as high as possible,
given the individually or socially optimal strategy. In a sense, this differs from Sections 2.2
and 2.3, where bigger was not necessarily better (on the contrary!).
Equality is
Possible
U,I
λO,I
e < λe
U,I
λO,I
e > λe
Rµ
C
Rµ
C
→∞
Figure 2.13 Equality of Individual rates in Observable vs. Unobservable M/M/1-systems
U
There are three cases up for discussion. If λO
e > λe , the optimal thing to do is to reveal
the length of the queue to customers before their decision to either join or balk. Conversely,
U
if λO
e < λe , an unobservable system is preferred. The service provider will be indifferent in
U
those particular points for which λO
e = λe . Figures 2.13 and 2.14 represent this profitmaximizing tactic as a function of Rµ
, for both individually and socially optimal customer
C
behaviour (adopted from [5]).
Equality is
Possible
O,S
λ e > λU,S
e
U,S
λO,S
e <λ e
Rµ
C
Rµ
C
→∞
Figure 2.14 Equality of Social rates in Observable vs. Unobservable M/M/1-systems
21
The employed tactic has to be modified according to the characteristics of the system under
consideration. We can, however, formulate three general principles to keep in mind. Their
mathematical proof is rather extensive and will therefore be omitted here. For those interested,
we refer to [5].
1. The range in which equality can occur, in terms of the relative value of the service, is
bigger under socially optimal customer behaviour.
2. Equality under individually optimal customer behaviour occurs for lower CR , i.e. for systems with a lower service gain, when compared to equivalent socially optimal behaviour.
3. In general, the incentive to reveal the queue length increases as the relative service gain
R
is low and the utilisation rate ρ is high.
C
We end this section with a remark on the situation in which both observable and unobservable
queue-joining rates are equal. In practice, service providers should not be indifferent as to
whether or not reveal the queue size. The most sensible decision is here to reveal queue length,
since this generates goodwill from customers. But that is a different field of study altogether...
22
Chapter 3
Multi-Server Systems with
Exponentially Distributed Interarrival
and Service Times (M/M/s)
3.1
Introduction
Now that results for a single-server system are in, it is time to have a look at what happens in
a multi-server environment, conducting the same analysis as in the previous chapter. At the
end of this chapter differences between both systems are explained and conclusions are drawn.
There is only one change in the list of assumptions of Section 2.2 that define this particular system [2], albeit an important one. More specifically, the first assumption becomes
1.
Poisson arrival stream of customers with rate λ, arriving at a service station with s
servers.
The changes that are made for the unobservable system remain the same as before.
The multi-server model considered here is in its essence nothing more than an extension of
the single-server case. Or vice versa, we can consider the M/M/1-system as a special case of
this more general M/M/s-system for which easier to use formulas can be derived. As a tool
for verifying the results found here, we can thus set s = 1 in the newly found equations and
compare with results from Chapter 2.
23
3.2
Observable System
3.2.1
Individual Behaviour
To determine the individually optimal threshold, we here need to make use of expression (1.2)
which takes into account the effect of multiple servers. Still to determine is the probability of
balking qn for an M/M/s/n-system. Birth-death analysis1 reveals that
qn =
with P0O

P O (sρ)s ρn−s
,n ≥ s
0
, n < s,
0
s!

s−1

P (sρ)k (sρ)s 1−ρn−s+1 −1


+ s!

k!
1−ρ
−1
= k=0
s−1
P sk ss



+ s! (n − s + 1)

k!
, ρ 6= 1
, ρ = 1.
k=0
P0O represents the probability that the M/M/s/n-system is empty. Determining the effective
queue-joining rate is straightforward, as this is given by the loss probability multiplied with the
system arrival rate λ. Using the above expression for qn , we get that
λO,I
= λ(1 − P0O
e
(sρ)s ni −s
ρ
),
s!
(3.1)
with ni = b Rsµ
c.
C
Please remark that the restriction of n ≥ s does not feature in this expression. Because
Rµ
≥ 1, the optimal threshold value ni will always satisfy this condition!
C
3.2.2
Social Behaviour
The expected number of customers in the system Ln is given by1
Ln =

sρ + P O (sρ)s
0
s + P O
0
s!
ρ
1−ρ
1−ρn−s
1−ρ
n−s+1
+ sρ
n−s
− nρ
(n−s)2 +n−s−2
, n ≥ s , ρ 6= 1
, n ≥ s , ρ = 1.
2
The socially optimal strategy ns is, following expression (1.3), the maximiser of
Gs = λR(1 −
(sρ)s n−s
P0O
ρ )
s!
− C sρ +
(sρ)
P0O
s
ρ
s! 1 − ρ
1 − ρn−s
+ sρn−s+1 − nρn−s
1−ρ
,
for n ≥ s and ρ 6= 1. The expression for Gs in the case of ρ = 1 can be stated in similar fashion.
1
Calculations were provided by co-promotor
24
Analogously to what we have encountered in the previous chapter, this expression is hard
to take derivatives of. It can be proved mathematically [6], however, that this function in n
is unimodal and that, similar as in single-server systems, ns ≤ ni . Of course nothing is taken
for granted, which is why we solved this expression for optimality in a numerical procedure in
Section 6.3.1.
The evolution of Gs as a function of n is depicted in Figure 3.1. This is very similar to
what was found before, showing one threshold value that maximises the net social gains.
250
Net Social Gains
200
150
100
50
0
0
3
6
9
12
15
18
21
24
27
30
33
36
39
42
45
48
51
54
57
60
Threshold value n
Figure 3.1 Evolution of net social gains in an Observable M/M/3-system, R = C = 10, λ = 8, µ = 10
The same conclusions can be made when the individually and socially optimal thresholds are
plotted for varying utilisation rates, as shown in Figure 3.2.
35
Threshold value n
30
25
20
Individual Threshold
15
Social Threshold
10
5
0
0
3
6
9 12 15 18 21 24 27 30 33 36 39 42 45 48 51 54 57 60
λ
Figure 3.2 Optimal Threshold values in an M/M/3-system,
25
R
C
= 1, µ = 10
Combining this knowledge with the expression for the loss probability qn , we get the following
equation for the socially optimal effective queue-joining rate:
λO,S
= λ(1 − P0O
e
(sρ)s ns −s
ρ
).
s!
(3.2)
To be able to compare the individual and social optimum, we make use of the numerical data
to generate Figures 3.3 and 3.4. First we let the system arrival rate λ vary, followed by the
. It is apparent that these graphs are similar to those derived
relative benefit of the service Rµ
C
in the previous chapter.
Effective Queue-joining rates
35
30
25
20
Individual Rate
15
Social Rate
10
5
0
0
3
6
9
12 15 18 21 24 27 30 33 36 39 42 45 48 51 54 57 60
λ
Figure 3.3 Individual vs. Social Optimisation in an Observable M/M/3-system,
R
C
= 1, µ = 10
≥ λO,S
The individual rate is always higher than or equal to the social rate (λO,I
e ), in accordance
e
with our previous findings. Individually optimal queueing behaviour will thus always be socially
suboptimal. For high utilisation rates, both effective rates converge to a value of 30 in this
λe
parameter setting. This situation corresponds to a server utilisation ρe (= sµ
) of 1.
Effective Queue-joining rates
26
24
22
20
18
Individual Rate
16
Social Rate
14
12
10
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
Rμ/C
Figure 3.4 Individual vs. Social Optimisation in an Observable M/M/3-system, ρ = 0.8, µ = 10
26
For the case in which the relative benefit of the service is so high that no customer wants to
balk (λe = λ), the rates converge to a value of 24. This is the multiplication of the number of
servers s with this λe .
3.2.3
Profit Maximisation
The options for letting customers take on the socially desired behaviour remain the same in a
multi-server environment. However, the number of servers, of course, has its effect on the size
of the admission fee or the waiting toll.
Poured into formulas, the admission fee a can be determined by the following expression for ns :
ns = b
(R − a)sµ
c.
C
Similarly, an appropriate waiting toll t can be derived from
ns = b
Rsµ
c.
C +t
The size of these payments depends on the specific values of R and C, and not solely on their
ratio. The cost of waiting per time unit C is therefore held constant in Figure 3.5. The
same general conclusions as in the previous chapter can be made. We do see, however, that
additional servers lead to a reduction in the variability of the payments. This is especially true
for the toll on waiting. Nevertheless, the size of the toll remains way higher than the equally
effective admission fee.
30
Size of fee/toll
25
20
15
Admission Fee
10
Waiting Toll
5
0
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19 20
Benefit of the Service R
Figure 3.5 Size of the Admission Fee/Waiting Toll in an Observable M/M/3-system, C = 10, µ = 10
27
3.3
Unobservable System
Only one parameter is missing to perform our analysis, more precisely the expected time in the
M/M/s-system Ts (λ). This expression can be readily found in literature on queueing [1].
Ts (λ) =
with
P0U
=
(sρ)s
1
+ P0U
,
µ
s!(1 − ρ)2 sµ
s−1
X
(sρ)k
k=0
(sρ)s
+
k!
s!(1 − ρ)
!−1
, ρ 6= 1.
This will be used in the following sections to determine the individually and socially optimal
queue-joining rate.
3.3.1
Individual Behaviour
Expression (1.4) should allow us to easily compute the optimal λ∗I . However, we face one
problem. The fact that the expression for P0U contains a sum, prohibits us to readily distill a
formula for the optimal value of λ. The same procedure as in some previous observable systems
is to be followed: for specific values of CR , µ and s, we will use iterations to find the optimal λ∗I
numerically (Section 6.3.2).
Putting this together with the cases in which a customer always balks (λe = 0) or always
joins (λe = λ), the effective queue-joining rate can only be formally stated as
λU,I
= min{λ∗I , λ}.
e
3.3.2
(3.3)
Social Behaviour
The same problem arises when trying to determine the socially optimal effective queue-joining
rate. The rather straightforward expression (1.6) is used, but it is again the Ts (λ) in an
M/M/s-system that complicates calculations. Therefore we also apply a numerical procedure
here (Section 6.3.3).
The socially optimal effective-queue joining rate is expressed by
λU,S
= min{λ∗ , λ}.
e
28
(3.4)
Figures 3.6 and 3.7 compare this optimum with the individual one found in the previous
section.
Effective Queue-joining rates
35
30
25
20
Individual Rate
15
Social Rate
10
5
0
0
3
6
9
12 15 18 21 24 27 30 33 36 39 42 45 48 51 54 57 60
λ
Figure 3.6 Individual vs. Social Optimisation in an Unobservable M/M/3-system,
R
C
= 1, µ = 10
is always higher or
The same conclusions as before can be drawn. The individual rate λU,I
e
equal to the social rate λU,S
e . As a difference to the observable case, however, we note that
both rates will never converge. The socially optimal balking point is at λ = 24.46, whereas it
is individually optimal to balk for higher utilisation rates λ = 28.96 (i.e. ρ = 0.97).
Effective Queue-joining rates
30
25
20
15
Individual Rate
Social Rate
10
5
0
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
Rμ/C
Figure 3.7 Individual vs. Social Optimisation in an Unobservable M/M/3-system, ρ = 0.8, µ = 10
Plotting the rates in function of Rµ
reveals that for services with high benefits, both rates
C
converge to a value of 24. This is similar to what we have found in Figure 3.4.
29
3.3.3
Profit Maximisation
Let us now have a look at how the service provider can maximise his revenues. As mentioned
before, he has the option to impose either an admission fee or a toll on waiting.
The size of the profit-maximising admission fee is given by
a = R − C.Ts (λ∗S ).
Analogously, the waiting toll is determined by
t=
R
− C.
Ts (λ∗S )
The lack of an expression for λ∗S induces us to solve this problem numerically. In Figure 3.8,
C is kept at a value of 10, µ = 10 and s = 3. The results again are very similar to what was
found in the single-server case.
70
60
Size of fee/toll
50
40
Admission Fee
30
Waiting Toll
20
10
0
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19 20
Benefit of the Service R
Figure 3.8 Size of the Admission Fee/Waiting Toll in an Unobservable M/M/3-system, C = 10, µ = 10
30
3.4
Observable vs. Unobservable Systems
In Chapter 2 we were able to derive a necessary and sufficient condition for equality of
observable and unobservable rates. Here, however, we are confronted with the problem of not
having an expression for the unobservable effective queue-joining rates. This makes it impossible
to derive this condition, inducing us to draw conclusions only based on the simulation data and
the resulting graphs.
3.4.1
Individual Behaviour
Figure 3.9 depicts the observable and unobservable rate in function of the system arrival rate
λ. The intersection between both rates lies just before λ = 30 (i.e. ρ = 1). We therefore have
no reason to doubt the expression derived in the single-server case, which states that 0 < ρi < 1.
For smaller utilisation rates, λU,I
≥ λO,I
e
e . After this critical point, the observable rate is
higher than the unobservable one, making it worthwhile to reveal queue length to customers.
Effective Queue-joining rates
35
30
25
20
Observable Rate
15
Unobservable Rate
10
5
0
0
3
6
9 12 15 18 21 24 27 30 33 36 39 42 45 48 51 54 57 60
λ
Figure 3.9 Individual Optimisation in Observable vs. Unobservable M/M/3-systems,
R
C
= 1, µ = 10
Making the relative benefit of the service variable, it is clear to see in Figure 3.10 that an
intersection point between both optimal rates cannot be readily distinguished. What we can
observe, is that both effective queue-joining rates converge to 24 for services with a high benefit. As mentioned before, this is because in this case no customer will balk, resulting in λe = λ.
For very low values of Rµ
, it is better to make a queue observable. High benefits imply that
C
an unobservable system gives better returns for the service provider. In between is a zone in
which it is not clear what the optimal tactic is (reveal queue length or not). Here it depends
on the specifics of the system parameters.
31
Effective Queue-joining rates
30
25
20
15
Observable Rate
Unobservable Rate
10
5
0
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
Rμ/C
Figure 3.10 Individual Optimisation in Observable vs. Unobservable M/M/3-systems, ρ = 0.8, µ = 10
3.4.2
Social Behaviour
The same comparison can also be done for socially optimal behaviour. This results in Figures
3.11 and 3.12. Again note that we cannot mathematically derive an expression for equality
and λU,S
of λO,S
e , since we have no equation regarding the unobservable rate.
e
Effective Queue-joining rates
35
30
25
20
Observable Rate
15
Unobservable Rate
10
5
0
0
3
6
9 12 15 18 21 24 27 30 33 36 39 42 45 48 51 54 57 60
λ
Figure 3.11 Social Optimisation in Observable vs. Unobservable M/M/3-systems,
R
C
= 1, µ = 10
The same conclusions as under individually optimal behaviour can be made. There exists an
intersection point before which the unobservable rate is greater than or equal to the observable
one. After this point, the observable rate is higher, indicating that it is beneficial to the service
provider to reveal information on the queue length to customers. Both rates do not converge,
since they make use of different sources of information.
32
Figure 3.12 compares the same rates, but now expressed in function of Rµ
. For very high
C
service gains, both rates converge. This is because customers will always decide to join, regardless of the fact whether they have information on the queue length or not. For this choice of
parameters, the resulting effective queue-joining rate is 24.
Effective Queue-joining rates
30
25
20
15
Observable Rate
Unobservable Rate
10
5
0
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
Rμ/C
Figure 3.12 Social Optimisation in Observable vs. Unobservable M/M/3-systems, ρ = 0.8, µ = 10
3.4.3
Profit Maximisation
In general, the single- and multi-server systems yield the same conclusions. And this comes
of course as no surprise, since the M/M/1-system is nothing more than a specific case of the
M/M/s-system discussed in this chapter. The tactics and remarks on revealing queue length
in Section 2.4.3 are thus still valid here.
33
3.5
Single-Server vs Multi-Server Systems
What this system does teach us, is the effect of the number of servers on the individual and
social rates in observable and unobservable systems. Particularly, it is interesting to investigate
on what an increase of the number of servers implies for the four effective queue-joining rates
we are considering.
This effect can be determined by plotting a bisector line that indicates an equally proportional increase in λe as in the number of servers s. In Figure 3.13 this is indicated with a thin
black line. The other curves show whether or not the calculated rates follow an increase in s
proportionately.
90
90
80
80
Effective Queue-joining rate
Effective Queue-joining rate
For both types of behaviour and both systems, it is apparent that a doubling in the number of servers implies at least a doubling in the effective queue-joining rates. In most cases, the
rate increases even more! This means that having a single server with a service rate of sµ is
suboptimal to having s servers, each with a service rate µ, all else being equal. It is hard to say
this comes as a surprise, since this outcome is a much supported finding in traditional queueing
analysis [1]. This is the main reason why, for example, big websites use multiple smaller servers
instead of one big server.
70
60
50
40
Ind Obs Rate
30
20
10
0
1
2
3
4
5
6
7
8
9
70
60
50
40
Ind Unobs Rate
30
20
10
0
10
1
2
3
4
5
6
7
8
9
10
Servers
90
90
80
80
Effective Queue-joining rate
Effective Queue-joining rate
Servers
70
60
50
40
Soc Obs Rate
30
20
10
0
70
60
50
40
Soc Unobs Rate
30
20
10
0
1
2
3
4
5
6
7
8
9
10
1
Servers
2
3
4
5
6
7
8
9
10
Servers
Figure 3.13 Effect of the number of servers on effective queue-joining rates,
34
R
C
= 1, ρ = 0.8, µ = 10
One could argue that it would be more fair to compare the base M/M/1/n-system with an
M/M/s/(n + s − 1)-environment, since this provides an equal amount of places in the queue.
However, the effect of the number of servers is already considered in the computation of the
optimal ni and ns , see equation (1.2). If we were to use the same value for n, thereby neglecting
the principles of the threshold strategy, this gap would be smaller. Figure 3.14 demonstrates
this under individually optimal behaviour by plotting the loss probabilities for several system
utilisation rates ρ for the M/M/1/10-, M/M/3/12- and M/M/3/30-system.
The analysis throughout this chapter has been based on the blue and green curves, because of
the threshold strategy logic discussed in Chapter 1. It is clear to see that indeed comparing
a threshold of n with n + s − 1 (blue vs. red) results in smaller differences in loss probability
because of equal queuing capacity. However, this is not relevant in the scope of this thesis,
where our analysis is steered by customer behaviour.
0.6
Loss probabilities
0.5
0.4
Loss probability for M/M/1/10
0.3
Loss probability for M/M/3/12
0.2
Loss Probability for M/M/3/30
0.1
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Utilisation ρ
Figure 3.14 Loss probabilities under Individual Optimisation,
35
R
C
= 1, µ = 10
Chapter 4
Single-Server Systems with
Hyperexponentially Distributed
Interarrival Times (H2/M/1)
4.1
Introduction
The third system up for discussion is one in which interarrival times follow a hyperexponential
distribution that is composed of two phases. Using this model, it is possible to determine the
effect of additional variation in the arrival process on several system parameters, whilst still
making use of the advantages of exponential distributions.
The why and how of this reasoning are explained first, followed by a similar analysis as the one
in previous chapters.
The hyperexponential distribution is composed of h parallel phases, that are each typified
by an exponentially distributed sojourn time. As to not complicate things too much, we will
set h = 2. The arrival rates in phases one and two will be named λ1 and λ2 respectively, and
the probability of a customer entering that particular phase will be α and 1 − α respectively.
At any time, one and only one customer is present in one of the phases. Figure 4.1 gives a
clear overview of this concept.
α
λ1
(1)
Infinite
Supply
Arrival
1− α
λ2
(2)
1 customer
Figure 4.1 Conceptual Drawing of Hyperexponential interarrival times
36
To reduce the number of variables we will express both λ1 and λ2 in terms of α and λ. More
precisely,

λ = 2αλ
1
λ = 2(1 − α)λ,
2
1
2
≤ α < 1.
This gives lead to defining two variables that constitute the respective utilisation rates, namely
ρ1 =
λ1
µ
and
λ2
.
µ
ρ2 =
For now this might seem an arbitrary choice, but the reason behind this will become clear
shortly. Note that for α = 21 , λ1 = λ2 = λ, and we essentially deal with exponential interarrival
times.
The time a customer is present in either phase is equal to 12 . This is due to the choice that we
made for α in the above arrival rates: a smaller probability to join a phase is compensated by
a longer interarrival time. For example, the probability that a customer is present in phase 1
is given by
1
α/λ1
= .
Pr[S = 1] =
α/λ1 + (1 − α)/λ2
2
The same calculation can be made for the second phase of the hyperexponential distribution.
We define the random variable τ that models an arbitrary interarrival time. For an exponential
distribution [1], the expected interarrival time E[τ ] and the squared coefficient of variation
(SCV) Cτ2 are given by
1
E[τ ] = ,
λ
Cτ2 = 1.
The parameters described above allow us to calculate the same system characteristics for the
hyperexponential model under consideration. Concretely, the expected interarrival time is
computed as follows:
1
1
1
E[τ ] = α
+ (1 − α)
= .
2αλ
2(1 − α)λ
λ
To calculate Cτ2 , we first need to know the variation of a hyperexponentially distributed variable
with the parameters chosen as above. This is given by
στ2 =
α2 + (1 − α)2
2α(1 − α)λ2
,
1
≤ α < 1.
2
As a result, the coefficient of variation is then calculated as
Cτ2 =
στ2
α2 + (1 − α)2
=
E[τ ]2
2α(1 − α)
37
,
1
≤ α < 1.
2
1
0.95
0.9
value of α
0.85
0.8
0.75
0.7
0.65
0.6
0.55
0.5
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
Squared Coefficient of Variation
Figure 4.2 Relation between the SCV and the probability of joining the first phase
For α = 12 , we can observe that, again, we have to do with an exponential distribution (Cτ2 = 1)
with system arrival rate λ. For α → 1, Cτ2 keeps on rising, indicating increasing variability.
At its extreme, λ1 = 2λ and λ2 = 0. This implies that the first phase results in exponential
arrivals with doubled intensity and the second phase would lead to infinitely long interarrival
times. Therefore, it is best to not choose too high a value for Cτ2 in simulation. The expected
interarrival time E[τ ] will remain the same, namely λ1 . Figure 4.2 shows the relation between
these two variables graphically.
Expressing α in function of Cτ2 allows us to set a higher variation (Cτ2 ≥ 1) whilst still being able to compare with the base exponential model discussed in Chapter 2.
1
α = (1 +
2
s
1−
Cτ2
2
).
+1
(4.1)
Knowing this, we can perform the a similar analysis of observable and unobservable systems as
has been done before.
4.2
Observable System
The threshold strategy used by customers implies we have to do with an H2 /M/1/n-system.
Traditional analysis of the system can then provide an expression for both the loss probability
qn and the expected number of customers in the system Ln . However, things are not as easy as
they look. The set of equations for the steady-state probabilities that come out of said analysis
cannot be readily solved mathematically. One option is therefore to use numerical procedures
to get to results for several settings of the system parameters.
38
As will become clear shortly, another possibility is to use results from systems with an unlimited queueing capacity (H/M/1) to get to these desired probabilities. This gives us the
benefit of having analytical results. We will address both options in the following paragraphs.
Yet we first need to define some more quantities:
N
random variable, indicating the number of customers in the system (N ≥ 0)
S
random variable, indicating phase in which an arriving customer is placed (S = 1, 2)
(N, S) state of the system
p(k, j) Pr[N = k, S = j] in the H2 /M/1/n-system, 0 ≤ k ≤ n, j = 1, 2
p̂(k, j) Pr[N = k, S = j] in the H2 /M/1-system, k ≥ 0, j = 1, 2
Table 4.1 Quantities used in the analysis of the H2 /M/1-and H2 /M/1/n-system
The analytical approach through the H2 /M/1-system and the numerical solution for the H2 /M/1/nsystem use these quantities in their analysis.
Numerical solution for H2 /M/1/n
The most straightforward way to tackle this problem is to have a look at the set of balance
equations that result from Birth-Death analysis. We are confronted with a set of 2(n + 1)
equations for 2(n + 1) variables1 :
n
P


(p(i, 1) + p(i, 2)) = 1



i=0











λ1 p(0, 1) = µp(1, 1)







(1)
(µ + (1 − α)λ1 )p(n, 1) = αλ1 p(n − 1, 1) + αλ2 p(n − 1, 2) + αλ2 p(n, 2)












λ1 p(k − 1, 1) + λ2 p(k − 1, 2) = µ(p(k, 1) + p(k, 2))
,1 ≤ k ≤ n










(µ + λ1 )p(k, 1) = αλ1 p(k − 1, 1) + αλ2 p(k − 1, 2) + µp(k + 1, 1)
, 1 ≤ k ≤ n − 1.
(2)
(3)
These equations can, however, not be distilled into a single expression for all p(k, j)’s. It is
thus not as straightforward as one would think. We need to resort to numerical procedures to
get solutions for the steady-state probabilities by solving this set of linear equations. Section
6.4.1 describes this method.
1
Calculations were provided by co-promotor
39
Once we do have numerical results, the expressions for the loss probability qn and expected
number of customers in the system Ln can be derived.
Concretely, the loss probability is given by
qn = 1 − ρ−1 (1 − p(0, 1) − p(0, 2)).
At its turn, the expected number of customers in the system can be expressed in terms of the
p(k, j)’s as
n
X
Ln =
k[p(k, 1) + p(k, 2)].
k=1
Analytical solution via H2 /M/1
A downside of the previously discussed solution method is that we do not have analytical formulas. For a change in parameter setting, the whole procedure would have to be repeated. It
is possible, however, to express the steady-state probabilities p(k, j) in a concise way using a
detour through the unlimited H2 /M/1-system.
When analysing the balance equations of this system, we observe that p(k, j) and p̂(k, j) satisfy
the same set of equations (1),(2) and (3), if we consider k ≤ n − 1 in (2) and k ≤ n − 2 in
(3). These equations can thus be used to express the couple (p(k, 1), p(k, 2)) as a function of
(p(l, 1), p(l, 2)), l < k, for decreasing values of k < n. Eventually, this leads to an expression
for (p(k, 1), p(k, 2)) in terms of the remaining probabilities (p(0, 1), p(0, 2)).
Since these balance equations are the same in the bounded and unbounded system, the expression of p̂(k, j) in terms of p̂(0, j) is formally the same as p(k, j) in terms of p(0, j). Therefore,
we can simply substitute p(k, j) and p(0, j) in these expressions.
For H2 /M/1 we find that
Q1 (z) =
in which Qj (z)=
ˆ
∞
P
T1 (z)
−D(z)
and
Q2 (z) =
T2 (z)
,
−D(z)
z k p̂(k, j), j = 1, 2. In these expressions, −D(z) is given by
k=0
−D(z) = (1 −
40
z
z
)(1 − ).
z1
z2
The unknown z1 and z2 are the poles of the Qj (z)’s. They are at their turn given by
z1,2 =
(Cτ2
1 + 2ρ ±
+ 1)
q
2
Cτ −1
1 − 4ρ(1 − ρ) C
2 +1
τ
4(Cτ2 ρ + ρ2 )
.
(4.2)
The numerators are defined as
"
# "
#"
#
T1 (z)
t11 (z) t12 (z) p̂(0, 1)
=
,
T2 (z)
t21 (z) t22 (z) p̂(0, 2)
in which
"
# "
#
t11 (z) t12 (z)
1 − (1 + ρ2 )z + (1 − α)ρ2 z 2
−αρ2 z 2
=
.
t21 (z) t22 (z)
−(1 − α)ρ1 z 2
1 − (1 + ρ1 )z + αρ1 z 2
Since we now have an expression for Qj (z) in terms of the p̂(0, j)’s, we can use this to create a solution for the steady-state probabilities p(k, j) of the system with limited capacity.
Concretely, by means of a partial fraction expansion, we get that

p(k, 1) =
p(k, 2) =
T1 (z1 ) −k
z
1−z1 /z2 1
T2 (z1 ) −k
z
1−z1 /z2 1
+
+
T1 (z2 ) −k
z ,
1−z2 /z1 2
T2 (z2 ) −k
z
1−z2 /z1 2
, 1 ≤ k ≤ n.
These expressions are also valid for k = n, as verified numerically.
The only thing left to determine are the expressions for p(0, j). Using the fact that the probability of being in either phase is 12 (i.e. Pr[S = j]), a set of two equations can be set up.

p(0, 1) 1 + t11 (z1 )f (z1 ) + t11 (z2 )f (z2 ) + p(0, 2) t12 (z1 )f (z1 ) +
1−z1 /z2
1−z2 /z1
1−z1 /z2
t
(z
)f
(z
)
t
(z
)f
(z
)
t
p(0, 1) 21 1 1 + 21 2 2 + p(0, 2) 1 + 22 (z1 )f (z1 ) +
1−z1 /z2
1−z2 /z1
1−z1 /z2
t12 (z2 )f (z2 )
1−z2 /z1
= 21 ,
t22 (z2 )f (z2 )
1−z2 /z1
= 21 ,
where f (z) is given by
f (z)=
ˆ
1 − z −n
.
z−1
This set of linear equations for p(0, 1) and p(0, 2) is easily solved and thus allows us to derive an
analytical expression for the expected number of customers in the system Ln . More precisely,
T1 (z1 ) + T2 (z1 )
Ln =
(z1 − 1)(1 − z1 /z2 )
T1 (z2 ) + T2 (z2 )
1 − z1n
1 − z2n
−n
−n
− nz1 +
− nz2
(z2 − 1)(1 − z2 /z1 ) 1 − z2−1
1 − z1−1
41
, ρ 6= 1.
Important to note is that these expressions are also true for ρ > 1, despite the fact that
they are based on results for ρ < 1. For ρ = 1, some more calculations show that
qn = 2n 2n+1 (α2 + (1 − α)2 + 2α(1 − α)n) − (1 − 2α)2
and
−1
α(1 − α)n(n + 1)2n+1 + (2α − 1)2 ((n − 1)2n + 1)
Ln =
2n+1 (α2 + (1 − α)2 + 2α(1 − α)n) − (1 − 2α)2
, ρ = 1,
, ρ = 1.
This analytical procedure yields exactly the same solutions for the effective queue-joining rates
as in the numerical method. The tables in Section 6.4.1 are thus equal to what is found using
these results.
4.2.1
Individual Behaviour
The individually optimal threshold value ni does not depend on the distribution of the interarrival times and is therefore given by equation (1.1). It is only dependent on the value of the
ratio of Rµ and C.
Given the expression for the loss probability in the previous section, we can state the individually optimal effective queue-joining rate as
λO,I
= λ[ρ−1 (1 − p(0, 1) − p(0, 2))],
e
(4.3)
c.
in which the values for p(0, j) are calculated for n = ni = b Rµ
C
4.2.2
Social Behaviour
Finding ns is not as straightforward. The net social gains per time unit are given by equation
(1.3). Filling in the newly found expressions for qn and Ln yields
−1
Gs = λR[ρ (1 − p(0, 1) − p(0, 2))] − C
n
X
k[p(k, 1) + p(k, 2)].
k=1
This is a very complicated equation to solve, which induces us once again to resort to numerical
procedures. Figures 4.3 and 4.4 show that the same general conclusions as before remain
true, namely that Gs has one maximum and that ns ≤ ni .
42
50
45
Net Social Gains
40
35
30
25
20
15
10
5
0
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
Threshold value n
Figure 4.3 Evolution of net social gains in an Observable H2 /M/1-system, R = C = 10, λ = 8, µ = 10,
Cτ2 = 4
12
Threshold value n
10
8
6
Individual Threshold
Social Threshold
4
2
0
0
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18 19 20
λ
Figure 4.4 Optimal Threshold values in an H2 /M/1-system,
R
C
= 1, µ = 10, Cτ2 = 4
Similar to the individual optimum, the socially optimal effective queue-joining rate can be
formally stated as
λO,S
= λ[ρ−1 (1 − p(0, 1) − p(0, 2))],
(4.4)
e
using ns in the calculations for p(0, j).
Analogously to the previous chapters, the effective queue-joining rates under both individually and socially optimal behaviour can now be compared. This is done in Figures 4.5 and
4.6, letting the system utilisation rate ρ or the relative benefit of the service CR vary respectively.
43
Effective Queue-joining rates
12
10
8
Individual Rate
6
Social Rate
4
2
0
0
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19 20
λ
Figure 4.5 Individual vs. Social Optimisation in an Observable H2 /M/1-system,
R
C
= 1, µ = 10, Cτ2 = 4
For very high system utilisation rates, both curves converge to an effective rate of 10, which is
equal to the service rate µ. This indicates a server utilisation ρe of 1. It is also apparent that
≤ λO,I
λO,S
e , as was always the case before. Again, individually optimal behaviour leads to a
e
socially suboptimal solution.
Effective Queue-joining rates
9
8
7
6
Individual Rate
5
Social Rate
4
3
2
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
Rμ/C
Figure 4.6 Individual vs. Social Optimisation in an Observable H2 /M/1-system, ρ = 0.8, µ = 10, Cτ2 = 4
If the relative benefit of the service is so high that no customer wants to balk (λe = λ), both
rates converge. For our choice of parameters, this means that the effective queue-joining rate
will be 8.
These are analogous findings as in Chapter 2, meaning that additional variation has little
effect on the general conduct of the effective rates. This is as predicted, since the expected
interarrival time E[τ ] is the same in both cases, namely λ1 . However, we can already note that,
44
for both figures, the convergence is slower than in the exponential case. The true effect of
variation will be elaborated on at the end of this chapter.
4.2.3
Profit Maximisation
Analogously to the previous chapters, admission fees and waiting tolls can be imposed to make
customers behave in the socially optimal way. Recall that for an observable system, this means
finding a value for the admission fee a such that
ns = b
(R − a)µ
c.
C
Similarly, the appropriate waiting toll t can be found by setting
ns = b
Rµ
c.
C +t
For Cτ2 = 4, µ = 10 and C = 10, results are given by Figure 4.7. This is very alike to the
graphs in Section 2.2.3. The effect of variation is reflected in the values for ns , which typically
have an equal or higher value than under exponentially distributed interarrival times with the
same parameter settings. Since the effect of the relative benefit of the service on ns is rather
small, the effect on the admission fee and waiting toll is minimal. For our parameter choice,
there is only a noticeable difference for really high values of R.
25
Size of fee/toll
20
15
Admission Fee
10
Waiting Toll
5
0
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19 20
Benefit of the Service R
Figure 4.7 Size of the Admission Fee/Waiting Toll in an Observable H2 /M/1-system, C = 10, µ = 10, Cτ2 = 4
45
4.3
4.3.1
Unobservable System
Individual Behaviour
There is only one parameter that we do not yet know in order to come to an individually optimal
strategy: the expected time in the system Ts (λ). Markovian analysis of the H2 /M/1-system
comes to the rescue2 , showing that this is given by
Ts (λ) =
z1
,
µ(z1 − 1)
This z1 corresponds to the one defined in expression (4.2) to analytically derive expressions
regarding the H2 /M/1/n-system. We can now fill this into the requirement for joining (1.4) so
that we can find the optimal joining rate λ∗I
C
z1
≤ R.
µ(z1 − 1)
The complicated nature of z1 does not allow us to solve for λ∗ . Section 6.4.2 summarises
some numerical results. What is then left for us to do, is to combine this with the pure balking
(λe = 0) and pure joining (λe = λ) strategy. Thus, the individually optimal queue-joining rate
can only be formally stated as
λU,I
= min(λ∗I , λ).
(4.5)
e
4.3.2
Social Behaviour
The same reasoning applies to the socially optimal behaviour. Making use of expression (1.6),
we are able to find λ∗S , which is the maximiser of
Gs = λ∗S [R − C
z1
].
µ(z1 − 1)
Again we will need to resort to numerical procedures to find the optimal values for several
parameter settings (Section 6.4.3). In terms of parameters, we can only say that the optimal
queue-joining rate is given by
λU,S
= min(λ∗S , λ).
(4.6)
e
2
Calculations were provided by co-promotor
46
Figures 4.8 and 4.9 depict both rates, holding either the relative benefit of the service CR or
the utilisation rate ρ fixed. Unsurprisingly, the individually optimal effective queue-joining rate
is always higher or equal to the socially optimal rate (λU,S
≤ λU,I
e
e ). The overall conclusions
made in the previous chapters still apply to this case.
Effective Queue-joining rates
9
8
7
6
5
4
Individual Rate
3
Social Rate
2
1
0
0
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19 20
λ
Figure 4.8 Individual vs. Social Optimisation in an Unobservable H2 /M/1-system,
R
C
= 1, µ = 10, Cτ2 = 4
Under socially optimal behaviour, there is no balking until the system utilisation rate ρ reaches
0.54. From that point on, customers will always balk. The equivalent individual balking point
is at a higher utilisation rate, namely 0.79. Both rates will therefore never converge once they
have passed the social optimum.
Effective Queue-joining rates
9
8
7
6
5
Individual Rate
4
Social Rate
3
2
1
0
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
Rµ/C
Figure 4.9 Individual vs. Social Optimisation in an Unobservable H2 /M/1-system, ρ = 0.8, µ = 10, Cτ2 = 4
Both rates will converge for systems with a very high relative benefit to a value of 8 for our
choice of parameters. This is logical, since in this case nobody wants to balk and the effective
queue-joining rate will assume the value of λ.
47
Again, these results are similar to what has been concluded in previous chapters. We refer
to Section 4.5 for more details about the effect of additional variation.
4.3.3
Profit Maximisation
Figure 4.10 plots appropriate admission fees and waiting tolls for several values of R. These
payments tempt customers to abandon their individually optimal strategy and adopt the socially desired one. In formulas, we can compute the admission fee a as
a = R − C.Ts (λ∗S ).
In similar fashion, the waiting toll t is given by
t=
R
− C.
Ts (λ∗S )
The effect of variation is much clearer than in the observable case. For all values of R, both
the admission fee and waiting toll assume lower values.
30
Size of fee/toll
25
20
15
Admission Fee
10
Waiting Toll
5
0
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19 20
Benefit of the Service R
Figure 4.10 Size of the Admission Fee/Waiting Toll in an Unobservable H2 /M/1-system, C = 10, µ = 10,
Cτ2 = 4
48
4.4
Observable vs. Unobservable Systems
We are confronted with the same problem as in the previous chapter, in the sense that we are
not equipped with equations to find the intersection between the observable and unobservable
optimum. We can therefore only make conclusions based on the numerical procedures and their
resulting graphs.
4.4.1
Individual Behaviour
The comparison of both systems under individually optimal behaviour is depicted in Figure
4.11, where the effective rates are plotted in terms of the system arrival rates. The figure is
very similar to previous findings.
However, the particularities caused by additional variation are different. The intersection
point for this parameter setting is at ρ = 0.91. Only for higher utilisation of the system,
it is worthwhile to disclose information on the queue length to customers. For the system with
exponentially distributed interarrival times, this was true for ρ > 0.98. At first glance, it therefore seems that additional variation induces service providers to make their queues observable
for lower utilisation rates! In addition, the gap between both rates is higher here than with
exponential arrivals. The decision to reveal or not reveal information on the queue length will
therefore have a bigger impact on revenue than before.
Effective Queue-joining rates
12
10
8
6
Observable Rate
Unobservable Rate
4
2
0
0
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18 19 20
λ
Figure 4.11 Individual Optimisation in Observable vs. Unobservable H2 /M/1-systems,
Cτ2 = 4
49
R
C
= 1, µ = 10,
Plotting the effective arrival rates in both systems in terms of CR (Figure 4.12) does not allow
for an equally straightforward interpretation. It is apparent, however, that both rates converge
for services with very high benefits. Similar to what we found in Chapter 2, there first is
a region where λO,I
> λU,I
e
e , followed by a region where equality is possible at several points.
Finally, for higher benefits, we see that λO,I
< λU,I
e
e . The general conclusions we made before
therefore still apply here.
9
Effective Queue-joining rates
8
7
6
5
4
Observable Rate
3
Unobservable Rate
2
1
0
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
Rμ/C
Figure 4.12 Individual Optimisation in Observable vs. Unobservable H2/M/1-systems, ρ = 0.8, µ = 10,
Cτ2 = 4
Even so, there is one distinguishable difference: the zone for equality occurs for higher benefits
compared to the exponential case. This means that revealing information on the queue length
becomes beneficial for services with higher relative value.
50
4.4.2
Social Behaviour
Do these conclusions also apply to the social optimum? Let’s find out! Figure 4.13 plots the
socially desired effective-queue joining rate as a function of λ. The intersection point lies at a
value of ρ = 0.64, whereas this was ρ = 0.77 in the case with Poisson arrivals. It turns out that
the results from the previous section also hold under socially optimal behaviour.
Effective Queue-joining rates
9
8
7
6
5
Observable Rate
4
Unobservable Rate
3
2
1
0
0
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18 19 20
λ
Figure 4.13 Social Optimisation in Observable vs. Unobservable H2 /M/1-systems,
R
C
= 1, µ = 10, Cτ2 = 4
From Figure 4.14 we can conclude that the findings regarding the relative benefit of the service
are also true under socially optimal behaviour.
Effective Queue-joining rates
8
7
6
5
4
Observable Rate
3
Unobservable Rate
2
1
0
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
Rμ/C
Figure 4.14 Social Optimisation in Observable vs. Unobservable H2 /M/1-systems, ρ = 0.8, µ = 10, Cτ2 = 4
51
4.4.3
Profit Maximisation
As the results of the previous section have shown, the overall tactics and remarks on revealing
queue length in Section 2.4.3 are still valid here. We can, however, add three more general
principles to the three we already had formulated. Additional variation induces us to conclude
that
4.
For systems that have to cope with higher variation in the interarrival times, it is
beneficial to start revealing the queue length at lower utilisation rates ρ.
5.
For systems that have to cope with higher variation in the interarrival times, the
incentive to reveal the queue length is present for higher relative service gains CR .
6.
Systems with higher variation in their interarrival times experience a bigger gap
between the optimal effective queue-joining rates in an observable versus an unobservable system. The decision whether or not to reveal the queue length will thus have a
bigger impact on the service provider’s profit.
4.5
Effect of Additional Variation
This is arguably the most important section of this chapter, since it provides a (partial) answer
to the third research question. By comparing results from the M/M/1-system with a similar
H2 /M/1-system, we are able to study the effect of additional variation in the arrival process
on several system parameters.
Let us first have a look at the impact of variation on the net social gains. Comparing Figures
2.1 and 4.3 quickly reveals that these gains are significantly lower when additional variation
comes into play. This implies that we have found evidence for the detrimental effect of variation on social welfare, which already is a widespread belief in other domains (e.g. quality
management [13], bullwhip effect in supply chains [8]). This result is thus not all surprising!
Concerning the effective arrival rates, Figures 4.15 to 4.18 show the effect of the squared
coefficient of variation Cτ2 on λe . Please recall that Cτ2 = 1 corresponds to exponentially distributed interarrival times. It is apparent that under both types of behaviour and both types
of systems, the effective arrival rates will always decrease with increasing variation.
As discussed in the introduction of this chapter, for infinitely high values of Cτ , only the
first phase of the hyperexponential distribution will be reached. Since half of the time a customer is present in either phase, this implies that the effective queue-joining rates should tend
to half their value for a very high SCV.
52
Figure 4.15 shows the effect of additional variation on the rates in an observable system.
The individually optimal rate goes to a value of λO,I
= 3.91, the social rate to λO,S
= 3.51.
e
e
The sudden drop in the latter is due to a change in the optimal value of ns from 4 to 3.
Effective Queue-joining rates
9
8
7
6
Individual Rate
Social Rate
5
4
3
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19 20
Squared Coefficient of Variation
Figure 4.15 Effect of Variation in the Interarrival Times in an Observable System,
R
C
= 1, ρ = 0.8, µ = 10
The corresponding situation in an unobservable system is displayed in Figure 4.16. For a very
= 4.5. Note that this is exactly half
volatile arrival process, the individual rate becomes λU,I
e
of the value of the λ∗I found in Chapter 2 and not of the truly optimal effective arrival rate,
which was 8. This is due to the game-theoretic reasoning we made for the cases in which all
customers balk (λe = 0) and in which all customers join (λe = λ). The socially optimal rate
= 3.42.
also tends to half its value, being λU,S
e
Effective Queue-joining rates
9
8
7
6
Individual Rate
5
Social Rate
4
3
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19 20
Squared Coefficient of Variation
Figure 4.16 Effect of Variation in the Interarrival Times in an Unobservable System,
53
R
C
= 1, ρ = 0.8, µ = 10
Comparing the observable and unobservable rates leads us to make another conclusion. Figures 4.17 and 4.18 show this situation under individually and socially optimal behaviour
respectively. It is clear that additional variation has a bigger effect on the unobservable rates.
Effective Queue-joining rates
8.5
8
7.5
7
Observable Rate
6.5
Unobservable Rate
6
5.5
5
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18 19 20
Squared Coefficient of Variation
Figure 4.17 Effect of Variation in the Interarrival Times under Individual Behaviour,
R
C
= 1, ρ = 0.8, µ = 10
In Figure 4.17, the observable and unobservable rate even have an intersection point (around
Cτ = 9). This means that for this level of variation under a utilisation rate of ρ = 0.8, it is
worthwhile to make the queue visible to customers, whereas in Figure 4.11 this is only true
for ρ > 0.91. For some higher value of Cτ2 , the rates will intersect again because the asymptotic
value for the observable rate is lower than for the unobservable one.
Figure 4.18 shows the same rates, but now under socially optimal behaviour. Again, the
unobservable rate is steeper than the observable one for increasing values of Cτ2 .
Effective Queue-joining rates
7.5
7
6.5
6
5.5
Observable Rate
5
Unobservable Rate
4.5
4
3.5
3
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18 19 20
Squared Coefficient of Variation
Figure 4.18 Effect of Variation in the Interarrival Times under Social Behaviour,
54
R
C
= 1, ρ = 0.8, µ = 10
Chapter 5
Single-Server Systems with
Hyperexponentially Distributed
Service Times (M/H2/1)
5.1
Introduction
The same hyperexponential distribution as in the previous chapter is used, albeit now to model
the service times of the system. The random variable x models the system’s service times.
Similar to the case with hyperexponentially distributed interarrival times, we define two service
rates as

µ = 2αλ
1
1
µ = 2(1 − α)λ,
≤ α < 1.
2
2
This gives lead to defining two variables that constitute the respective utilisation rates, namely
ρ1 =
λ
µ1
and
ρ2 =
λ
.
µ2
If we compute the expected service time E[x] and squared coefficient of variation Cx2 , we notice
again that we are dealing with a system that has the same expected service times, but with
more variation around this value. The parameter α allows us to increase or decrease variation
in the system. For α = 21 (or SCV Cx2 = 1), we again deal with an exponentially distributed
variable.
The definition of the system considered here is thus very similar to the one discussed in Chapter 4. What makes the analysis a little more complex here, is the fact that there is not always
exactly one customer present in one of the phases of the hyperexponential distribution. This
is only true if the system is not empty!
55
The goal of this chapter is to check whether or not the effect of additional variation in the
service rate has the same impact as with the interarrival times. The general structure of the
analysis will thus follow the same principles as in the previous chapter.
5.2
Observable System
In order to model the observable system, we face the same problem as in the previous chapter. The M/H2 /1/n-system provides us with a set of balance equations that cannot be solved
mathematically to fit into a single expression for qn or Ln . Again we will consider the option
of determining the value of the steady-state probabilities through the infinite M/H2 /1-system.
The nature of this system requires us to first define some more parameters:
random variable, indicating the number of customers in the system (N ≥ 0)
random variable, indicating the phase in which a served customer is placed
S = 1, if a customer is in phase 1 and N > 0
S = 2, if a customer is in phase 2 and N > 0
(N, S) state of the system
p(0)
Pr[N = 0] in the H2 /M/1/n-system
p(k, j) Pr[N = k, S = j] in the H2 /M/1/n-system, 1 ≤ k ≤ n, j = 1, 2
p̂(0)
Pr[N = 0] in the M/H2 /1-system
p̂(k, j) Pr[N = k, S = j] in the H2 /M/1-system, k ≥ 1, j = 1, 2
N
S
Table 5.1 Quantities used in the analysis of the M/H2 /1-and M/H2 /1/n-system
We will now use these to construct a numerical solution of the M/H2 /1/n-system and an analytical solution via the M/H2 /1-system, similar to the case with hyperexponentially distributed
interarrival times.
56
Numerical solution for M/H2 /1/n
We will again start off with analysing the finite M/H2 /1/n-system through its set of balance
equations. Queueing analysis yields a set of 2n + 1 equations for 2n + 1 variables1 . However,
they cannot be readily reduced to a single expression for the system’s parameters. Notice that
we are dealing with one variable less than in the previous chapter. This is because the probability of an empty system is captured in a single variable p(0) instead of two p(0, j)’s.
More precisely,

n
P


p(0)
+
(p(i, 1) + p(i, 2)) = 1



i=0











λp(0) = µ1 p(1, 1) + µ2 p(1, 2)












(λ + µ1 )p(1, 1) = αλp(0) + αµ1 p(2, 1) + αµ2 p(2, 2)
(1)






µ1 p(n, 1) = λp(n − 1, 1)












µ1 p(k, 1) + µ2 p(k, 2) = λ(p(k − 1, 1) + p(k − 1, 2))
,2 ≤ k ≤ n










(λ + µ1 )p(k, 1) = λp(k − 1, 1) + αµ1 p(k + 1, 1) + αµ2 p(k + 1, 2)
, 2 ≤ k ≤ n − 1.
(2)
(3)
Solving this set of equations numerically provides us with values for the steady-state probabilities. Section 6.5.1 gives an overview for several parameter choices.
The loss probability qn can easily be computed as
qn = 1 − ρ−1 (1 − p(0)).
To get to an expression for Ln , we need to resort to the general definition of the expected
number of customers in the system, which says that
Ln =
n
X
k(p(k, 1) + p(k, 2)).
k=1
These two variables are, as always, the basis of the analysis of individual and social behaviour
in observable systems.
1
Calculations were provided by co-promotor
57
Analytical solution via M/H2 /1
As in the previous chapter, we prefer to have analytical expressions to work with. Analysis of
the balance equations of the unbound M/H2 /1-system reveals that a number of them are equal
to those found in the previous M/H2 /1/n-section.
More precisely, equations (1),(2) and (3) are the same, given we consider k ≤ n − 1 in (2)
and k ≤ n − 2 in (3). When we apply the same logic as in Chapter 4, we see that the couple
(p̂(k, 1), p̂(k, 2)) can be expressed in terms of p̂(0). Since the same expression is true for the
p(k, j)’s, we can derive this expression for the infinite system and simply substitute p(k, j) and
p(0).
For M/H2 /1 we find that
Q1 (z) = z
T1 (z)
p̂(0)
D(z)
and
where Qj (z)=
ˆ
∞
X
Q2 (z) = z
T2 (z)
p̂(0),
D(z)
z k p̂(k, j), j = 1, 2.
k=1
The denominator D(z) is given by
D(z) = (1 −
z
z
)(1 − ),
z1
z2
in which z1 and z2 are thus the poles of the Qj (z)’s, given by
"
z1,2 = ρ
−1
ρ
1+ ±
2
s
#
ρ2
Cx2 − 1
+ (1 − ρ) 2
.
4
Cx + 1
T1 (z) and T2 (z) incorporate system-specific parameters in such a way that

T (z) =
1
T (z) =
2
αρ1 (1+ρ2 −ρ2 z)
1+(1−α)ρ1 +αρ2
(1−α)ρ2 (1+ρ1 −ρ1 z)
.
1+(1−α)ρ1 +αρ2
Analogously to this, we can express p(k, 1) and p(k, 2) in terms of p(0) for the M/H2 /1/n-system
as follows

p(k, 1) = T1 (z1 )p(0) z −(k−1) + T1 (z2 )p(0) z −(k−1)
1−z1 /z2 1
1−z2 /z1 2
−(k−1)
−(k−1)
T
(z
)p(0)
T
p(k, 2) = 2 1
z
+ 2 (z2 )p(0) z
, 1 ≤ k ≤ n − 1.
1−z1 /z2
1
1−z2 /z1
58
2
For the probabilities at the threshold value (i.e. for which k = n), we know from the state
diagram that

p(n, 1) = ρ p(n − 1, 1)
1
p(n, 2) = ρ p(n − 1, 2).
2
The probability of an empty system can be found using the property that, given that the system
is not empty, the time a customer is present in either phase is divided equally over the phases
(Pr[S = j|N > 0] = 21 ). We can state that
n
X
1
p(k, j) = (1 − p(0)).
2
k=1
These expressions for the steady-state probabilities p(k, j) allow us to redefine the expression
for the expected number of customers in the system Ln . Concretely,
T1 (z1 ) + T2 (z1 )
Ln =
(1 − z1 /z2 )(1 − 1/z1 )
1 − z1−n
−(n−1)
− nz1
p(0)
1 − z1−1
T1 (z2 ) + T2 (z2 )
+
(1 − z2 /z1 )(1 − 1/z2 )
1 − z2−n
−(n−1)
p(0)
− nz2
1 − z2−1
, ρ 6= 1.
+ n[p(n, 1) + p(n, 2)]
Although the formulas for the infinite M/H2 /1-system only apply for ρ < 1, the above expressions for qn and Ln are also valid for bigger values of the system utilisation rate.
For ρ = 1, the expression for the loss probability is given by
n
1
qn = p(0) = (1 + 2(2α − 1) (1 −
) + 4nα(1 − α))−1
2
2
, ρ = 1.
Similarly, the expected number of customers in the system can be expressed as
n 1
2
)
Ln = p(0) n + 2α(1 − α)n(n − 1) + 2(1 − 2α) (1 − (n + 1)
2
59
, ρ = 1.
5.2.1
Individual Behaviour
The optimal threshold value under individual behaviour is not affected by the type of distribution the interarrival and service times take on. This ni is given by equation (1.1).
As we did before, the effective queue-joining rate is determined by multiplying the system
arrival rate λ with the probability of joining, i.e. 1 − qn . This results in having
λO,I
= µ(1 − p(0)),
e
(5.1)
c in all computations for p(0).
where n = ni = b Rµ
C
5.2.2
Social Behaviour
To determine the socially optimal threshold ns we have to take a detour through equation (1.3),
which expresses the net social gains Gs as a function of n. Using the above expressions for qn
and Ln , we find that
n
X
k(p(k, 1) + p(k, 2))].
Gs = λR[ρ−1 (1 − p(0))] − C[
k=1
Figures 5.1 and 5.2 show that earlier conclusions concerning the unimodal nature of this
function remain valid in this system. The statement that ns ≤ ni can also be made here. The
expression for the socially optimal effective queue-joining rate looks equal to the individual
case, albeit that the computations for p(0) include n = ns . Formally, this rate is stated as
λO,S
= µ(1 − p(0)).
e
(5.2)
60
Net Social Gains
50
40
30
20
10
0
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
Threshold value n
Figure 5.1 Evolution of net social gains in an Observable M/H2 /1-system, R = C = 10, λ = 8, µ = 10,
Cx2 = 4
60
Once again it is clear that additional variation results in lower social welfare. If we compare
Figure 4.3 with the one here, we see that variation is less detrimental to the gains when it
occurs in the service rates. The optimal gains here are higher than those found in the previous
chapter.
12
Threshold value n
10
8
6
Individual Threshold
Social Threshold
4
2
0
0
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18 19 20
λ
Figure 5.2 Optimal Threshold values in an M/H2 /1-system,
R
C
= 1, µ = 10, Cx2 = 4
Knowing all this, we can start comparing the individually and socially optimal rates. In Figure
5.3, they are plotted as a function of the system arrival rate λ, whereas the relation with the
relative benefit of the service CR is looked at in Figure 5.4. The same general conclusions as
before apply. The social rate is always lower than or equal to the individual one (λO,S
≤ λO,I
e
e ).
The effective rates here are lower than in the exponential case discussed in Chapter 2. They
are, however, a little higher than in the system with hyperexponentially distributed interarrival
times. This adds to our belief that variation in the service rates has slightly less of a negative
effect on the optimal effective queue-joining rates than was the case with a highly variable
arrival process. Nonetheless, the general effect is much the same.
Effective Queue-joining rates
12
10
8
6
Individual Rate
4
Social Rate
2
0
0
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19 20
λ
Figure 5.3 Individual vs. Social Optimisation in an Observable M/H2 /1-system,
61
R
C
= 1, µ = 10, Cx2 = 4
Effective Queue-joining rates
9
8
7
6
Individual Rate
5
Social Rate
4
3
2
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
Rμ/C
Figure 5.4 Individual vs. Social Optimisation in an Observable M/H2 /1-system, ρ = 0.8, µ = 10, Cx2 = 4
5.2.3
Profit Maximisation
As we did in all foregoing chapters, we want to quantify an admission fee or a waiting toll
that would induce customers to adopt socially desired behaviour. For our choice of parameters,
the results are given by Figure 5.5. The interpretation of this graph is identical to previous
chapters. An admission fee is preferred since it shows lower variability and implies a smaller
transaction between customers and the service provider.
35
Size of fee/toll
30
25
20
Admission Fee
15
Waiting Toll
10
5
0
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19 20
Benefit of the Service R
Figure 5.5 Size of the Admission Fee/Waiting Toll in an Observable M/H2 /1-system, C = 10, µ = 10, Cx2 = 4
62
5.3
Unobservable System
When the customer has no information on the queue length, his joining/balking decision will
be based on a predefined chance. There is therefore no threshold that complicates our analysis.
We are simply dealing with an M/H2 /1-system.
The only parameter that we do not know yet is the expected time in the system Ts (λ). Using
the Pollaczek-Khintchine equation for M/G/1-systems [4][12], we get that
2
C −1
1 1 + x2 ρ
.
Ts (λ) = +
µ
1−ρ
(5.3)
All formulas derived in this section are thus valid for more general distributions of the service
rates than the hyperexponential one!
5.3.1
Individual Behaviour
Using the requirement for joining given by (1.4), we can find the optimal queue-joining rate λ∗I
by solving
!
C 2 −1
1 1 + x2 ρ
C
+
≤ R.
µ
1−ρ
This is solved for λ in Section 6.5.2. We can rewrite this expression to get to λ∗I immediately.
More precisely,
C
1 − Rµ
λ∗I = µ
.
2
C
1 + Cx2−1 Rµ
It is clear that, for Cx2 > 1, this λ∗I will always be smaller than the one derived in Chapter
2. Combining this with the pure balking (λe = 0) and pure joining strategy (λe = λ), we can
state that
λU,I
= min(λ∗I , λ).
(5.4)
e
5.3.2
Social Behaviour
The same procedure can be repeated for the socially optimal rate. More precisely, we want to
maximise expression (1.6) given the Ts (λ) stated above. The net social gains per time unit are
thus given by
!
C 2 −1
1 1 + x2 ρ
+
].
Gs = λ[R − C
µ
1−ρ
63
Using the iterative procedure explained in Section 6.5.3, we find the optimal λ∗S . We can also
do this here by taking the derivative for λ and equal the expression to zero. This results in

s
λ∗S = µ 1 −
v

u
Cx2 −1
u
1
+
C t
2

2 −1
C
C
x
Rµ 1 +
2 Rµ
Clearly this expression is always smaller than or equal to (for Cx2 = 1) the λ∗S derived for the
base M/M/1-system. The combination of all three strategies is then stated as
λU,S
= min(λ∗S , λ).
e
(5.5)
Figure 5.6 compares the individual and social effective joining rate for different utilisation
rates. It is apparent that the same conclusions as before can be made. The social rate is always
equal to or lower than the individually optimal rate (λO,S
≤ λO,I
e
e ).
Effective Queue-joining rates
9
8
7
6
5
Individual Rate
4
Social Rate
3
2
1
0
0
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19 20
λ
Figure 5.6 Individual vs. Social Optimisation in an Unobservable M/H2 /1-system,
R
C
= 1, µ = 10, Cx2 = 4
Under individually optimal behaviour and under this parameter setting, customers balk if
ρ > 0.78. This critical balking point is way lower for socially desired behaviour, where it is at
ρ = 0.53. These results are similar to the case with hyperexponential interarrival times, but
lay at way lower utilisation rates than those discussed in Figure 2.6.
Figure 5.7 shows the same rates, albeit now plotted as a function of Rµ
. The same conC
clusions apply. For a more detailed explanation of the effect of variation in the service rates, we
refer to Section 5.5. Comparing these figures with Figures 4.8 and 4.9 reveal that the rates
in the unobservable system are lower here than in the case with hyperexponentially distributed
arrivals. The opposite was true in the observable system! An explanation for this will also be
given at the end of the chapter.
64
Effective Queue-joining rates
9
8
7
6
5
4
Individual Rate
3
Social Rate
2
1
0
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
Rμ/C
Figure 5.7 Individual vs. Social Optimisation in an Unobservable M/H2 /1-system, ρ = 0.8, µ = 10, Cx2 = 4
5.3.3
Profit Maximisation
Equally to the observable case, it is useful to know the size of an admission fee or waiting toll
that stimulates socially desired behaviour. The result is depicted in Figure 5.8. The nature
of the strategies used in unobservable systems lead to a smoother course of these curves. The
size of the waiting toll is higher than a similar admission fee.
30
Size of fee/toll
25
20
15
Admission Fee
10
Waiting Toll
5
0
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19 20
Benefit of the Service R
Figure 5.8 Size of the Admission Fee/Waiting Toll in an Unobservable M/H2 /1-system, C = 10, µ = 10,
Cx2 = 4
Comparing the size of these welfare-optimising payments with those in Figure 2.8, reveals
that they are considerably lower in this case. This is due to the lower effective queue-joining
rates that result from additional variation.
65
5.4
Observable vs. Unobservable Systems
Again we do not have analytical expressions that would allow us to readily compute the intersection between the optimal queue-joining rates in observable and unobservable systems. Our
conclusions will thus be based merely on graphical interpretation and results from the numerical
procedures in Chapter 6.
5.4.1
Individual Behaviour
Figures 5.9 and 5.10 demonstrate the comparison between both rates under individually
optimal behaviour. In terms of utilisation rate, the rates intersect around a value of ρ = 0.88.
This is a little lower than in the H2 /M/1-system, but a lot less than the utilisation rate of
98% in the base M/M/1-system. It is thus beneficial for service providers to reveal information
on the queue length to customers for lower utilisation rates when they are confronted with
additional variation in their service times. The same is true for hyperexponentially distributed
interarrival times, as discussed in the previous chapter.
Effective Queue-joining rates
12
10
8
6
Observable Rate
Unobservable Rate
4
2
0
0
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18 19 20
λ
Figure 5.9 Individual Optimisation in Observable vs. Unobservable M/H2 /1-systems,
R
C
= 1, µ = 10, Cx2 = 4
Figure 5.10 shows the same comparison for varying values of the relative benefit of the service.
We notice that the zone in which both rates have intersection points is situated at higher
benefits. Compared to Figure 2.10, it is clear that making the queue observable will be
beneficial for higher values of CR .
66
9
Effective Queue-joining rates
8
7
6
5
4
Observable Rate
3
Unobservable Rate
2
1
0
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
Rμ/C
Figure 5.10 Individual Optimisation in Observable vs. Unobservable M/H2/1-systems, ρ = 0.8, µ = 10,
Cx2 = 4
5.4.2
Social Behaviour
Now let us do the same for observable and unobservable rates under socially optimal behaviour.
Figure 5.11 analyses these rates in terms of the system utilisation rate ρ. The ”beneficial-tomake-observable-point” is located around ρ = 0.62. This is considerably lower than what we
found in the case with Poisson arrivals. Therefore, we daresay that the conclusions made in
the previous section remain valid under social behaviour.
Effective Queue-joining rates
9
8
7
6
5
4
Observable Rate
3
Unobservable Rate
2
1
0
0
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18 19 20
λ
Figure 5.11 Social Optimisation in Observable vs. Unobservable M/H2 /1-systems,
R
C
= 1, µ = 10, Cx2 = 4
Our findings regarding the comparison as a function of the relative benefit of the service also
remain valid, as is shown in Figure 5.12. The zone of equality is situated around higher
valuable services.
67
Effective Queue-joining rates
8
7
6
5
4
Observable Rate
3
Unobservable Rate
2
1
0
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
Rμ/C
Figure 5.12 Social Optimisation in Observable vs. Unobservable M/H2 /1-systems, ρ = 0.8, µ = 10, Cx2 = 4
5.4.3
Profit Maximisation
As is made clear throughout the analysis, the effect of variation in the service rates is comparable to when it occurs in the arrival process. The principles formulated in Section 4.4.3 can
therefore be generalised to
4.
For systems that have to cope with higher variation in the interarrival or service times,
it is beneficial to start revealing the queue length at lower utilisation rates ρ.
5.
For systems that have to cope with higher variation in the interarrival or service times,
the incentive to reveal the queue length is present for higher relative service gains CR .
6.
Systems with higher variation in their interarrival or service times experience a bigger gap
between the optimal effective queue-joining rates in an observable versus an unobservable
system. The decision whether or not to reveal the queue length will thus have a bigger
impact on the service provider’s profit.
It goes without saying that principles 1 to 3 from Section 2.4.3 still apply here.
68
5.5
Effect of Additional Variation
The detrimental effect of variation in the service times has been proven throughout this chapter. It was also discussed that their effect is much the same as when variation occurred in the
arrival process. We have created similar figures as in Section 4.5 to demonstrate the effect of
increasing variation on the effective queue-joining rates in observable and unobservable systems.
In order to explain to where the effective rates tend for very high values of Cx2 , we need to
split up between the observable and unobservable case.
The observable rates will decrease, but they will not go to as low as half their value as was
the case in the previous chapter. The reason for this is that the probability that a customer
is present in phase j is defined here as Pr[S = j|N > 0] = 12 . This conditional probability is
bigger than the unconditional one since p(0) (i.e. Pr[N = 0]) is positive.
Figure 5.13 supports this reasoning by showing that the individual and social rates are relatively flat for high values of the squared coefficient of variation Cx2 .
Effective Queue-joining rates
9
8
7
6
Individual Rate
Social Rate
5
4
3
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19 20
Squared Coefficient of Variation
Figure 5.13 Effect of Variation in the Service Times in an Observable System,
R
C
= 1, ρ = 0.8, µ = 10
As is made clear in Figure 5.14, the unobservable rates do not follow the same logic. In
fact, for α → 1, they are zero! This can be explained by use of expression (5.3), which shows
the relation between Cx2 and the expected time in the system Ts (λ). It is clear that very
high variation leads to a very long expected time in the system. If we combine this with the
requirement for joining (1.4), the benefit of the service will never outweigh the costs of waiting
for an infinitely long time in the system. In order to provide a sensible service (i.e. one in
which Rµ
> 1), λe ultimately has to decrease to zero.
C
69
Effective Queue-joining rates
9
8
7
6
Individual Rate
5
Social Rate
4
3
2
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19 20
Squared Coefficient of Variation
Figure 5.14 Effect of Variation in the Service Times in an Unobservable System,
R
C
= 1, ρ = 0.8, µ = 10
This difference in course of the observable and unobservable rates is highlighted by plotting
them together in a single graph, as shown in Figures 5.15 and 5.16, under individually and
socially optimal behaviour respectively. These results support our decision to not choose a
coefficient of variation that is extremely high in the standard cases of the analysis. This could
have caused wrong interpretations.
Effective Queue-joining rates
8.5
8
7.5
7
6.5
Observable Rate
6
Unobservable Rate
5.5
5
4.5
4
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18 19 20
Squared Coefficient of Variation
Figure 5.15 Effect of Variation in the Service Times under Individual Behaviour,
70
R
C
= 1, ρ = 0.8, µ = 10
Effective Queue-joining rates
7.5
6.5
5.5
4.5
Observable Rate
Unobservable Rate
3.5
2.5
1.5
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18 19 20
Squared Coefficient of Variation
Figure 5.16 Effect of Variation in the Service Times under Social Behaviour,
R
C
= 1, ρ = 0.8, µ = 10
Consider the situation in which one can choose to either experience high variation in the arrival
process or in the service unit. As our analysis has shown, this choice depends on the type of
system. In an observable system, one should prefer it to occur in the service rates (as the lesser
of two evils, that is). Vice versa, in an unobservable environment, variation in the arrival rates
should be chosen.
However, as we have demonstrated throughout, a service provider should simply strive to
minimise variation as much as possible, as well in his service as in his arrival process. Shewhart
and Deming [13] knew that already in the 1930s...
71
Chapter 6
Numerical Procedures
6.1
Introduction
Throughout our analysis we have encountered problems with the analytical derivation of equations. Sometimes we were not able to find a mathematical expression for certain quantities or
we could not easily take the derivatives of a function to find its maximum value. This is where
several numerical procedures have come into play. We have used four different approaches, each
specific to a problem, that all fall under the term ”numerical procedure”. We will first describe
each of these methods, after which the concrete results are given for several systems. This is
the data on which all figures are based and thus can be considered to be the foundation for this
entire thesis.
6.1.1
Social Optimisation in an Observable System
The first analytical problem we faced had to do with finding the socially optimal threshold
value ns in observable systems. Equation (1.3) determines the specific expression for the net
social gains per time unit Gs that have to be maximised. Given the complicated nature of this
expression in n, it is very hard to take the derivative hereof and set it equal to zero.
To overcome this problem, an iterative search for ns is constructed. Starting at a value of
n = 1, the corresponding gains are calculated. This value for Gs is then compared with its
value in the previous iteration. When Gs (n) ≥ Gs (n − 1), the next iteration is started. This
goes on until we find a value for n where this test is no longer true. The optimum is consequently given by Gs (n − 1). Since the net social gains are unimodal in n [9], we are sure that
there is no other maximum besides the one found using this procedure.
This iterative method is programmed in Java and results in a list of values for ns , under
several parameter settings. This way we can determine the effect of the utilisation rate ρ and
the relative benefit of the service CR on the effective queue-joining rate λO,S
e .
72
6.1.2
Individual Optimisation in an Unobservable System
A second problem is of a completely different nature. In unobservable systems, we are faced with
the task of determining the unrestricted critical λ∗I that solves the equality given by equation
(1.4). In the basic M/M/1-system, this is rather straightforward. However, as the expression
for Ts (λ) becomes more complex, we need to resort to numerical procedures to find the optimal
joining rate.
Therefore, we have set up an iterative search for this optimal value λ∗I . Beginning at λ = 0
and increasing with 0.0001 every iteration, the equality is checked for the incumbent value of
λ. The properties of this procedure require us to set a precision boundary around the value
we wish to attain, since λ is increased stepwise. We choose to set an error margin of 10−4
for computational reasons. Concretely, this means that our objective needs to be comprised
between 99.99% and 100.01% of the targeted value. When a λ is encountered that fulfills this
condition, the search is stopped.
This numerical procedure is programmed in Java and results in a list of values for λ∗I , under several parameter settings. This again allows us to study the effect of CR on the effective
queue-joining rate λU,I
e .
6.1.3
Social Optimisation in an Unobservable System
This third situation is in fact nothing more than a combination of the first two. We are confronted with an expression for Gs (1.6) that needs to be maximised by letting λ vary. We
therefore iteratively search for λ∗S by comparing the Gs of each iteration with its value in the
previous round.
Concretely, we start the iterative procedure by setting λ = 0 and computing Gs (λ). For
every increase in λ by 0.0001 (i.e. precision of 10−4 ), we again compute the net social gains per
time unit Gs . When Gs (λ) ≥ Gs (λ − 0.0001), the next iteration is initiated. The optimum is
attained when this requirement is no longer fulfilled. The optimal net social gains are given by
Gs (λ − 0.0001).
One could argue that this procedure is equal to the one discussed in Section 6.1.1. This
is not true, since the variable we iterate on here is continuous, whereas the threshold n only
takes on integer values. That is why we need the precision margin in this case.
Once again, we have implemented this procedure using the Java programming language. The
constructed program offers us a list with values for λ∗S , under several parameter settings, allowing us to determine the effect of CR on the effective queue-joining rate λU,S
e .
73
6.1.4
Numerical Solution for a Set of Balance Equations
A totally different approach is needed to solve the set of balance equations that come from the
hyperexponential systems with limited queueing capacity (H2 /M/1/n and M/H2 /1/n). Using
my background as a Business Engineering Student, I decided to tackle this problem using solutions from the field of Operations Research.
More precisely, I modeled the set of equations as a linear programming model. The goal is
to let the sum of all probabilities be as close to 1 as possible, which constitutes the normalizing
condition. All other equations act as restrictions to this goal. As a result, we get a set of values
for the steady-state probabilities p(k, j) (and p(0)).
As an example, the set of equations in Section 4.2 can be rewritten as
Objective Function:
Min
n
X
(p(i, 1)+p(i, 2))−1,
i=0
subject to
n
X
(p(i, 1) + p(i, 2)) − 1 ≥ 0,
i=0
λ1 p(0, 1) = µp(1, 1),
(µ + (1 − α)λ1 )p(n, 1) = αλ1 p(n − 1, 1) + αλ2 p(n − 1, 2) + αλ2 p(n, 2),
λ1 p(k − 1, 1) + λ2 p(k − 1, 2) = µ(p(k, 1) + p(k, 2)), 1 ≤ k ≤ n,
(µ+λ1 )p(k, 1) = αλ1 p(k−1, 1)+αλ2 p(k−1, 2)+µp(k+1, 1), 1 ≤ k ≤ n−1,
p(k, j) ≥ 0.
Since the problem is made rather straightforward this way, I used the MS Excel Solver add-in
to calculate results. However, for every change in parameters, the program has to be executed.
This is the main reason why an analytical approach has been set up in Chapters 4 and 5.
74
6.2
Single-Server Systems with Exponentially Distributed
Interarrival and Service Times (M/M/1)
6.2.1
Social Optimisation in an Observable M/M/1/n-system
In Section 2.2.2 an expression is stated from which the socially optimal threshold value in
an observable M/M/1/n-system can be derived. We make use of the numerical procedure discussed in Section 6.1.1 to get results.
Table 6.1 summarises some outcomes for the optimal threshold ns , for different values of
= 10.
ρ, while holding Rµ
C
Table 6.1
Utilisation ρ
ns
0.1
9
0.2
8
Analogously, it is also possible to let
6.2, with ρ = 0.8 and µ = 10.
Table 6.2
Rµ/C
ns
1
1
2
1
3
2
4
2
0.3
7
Rµ
C
0.4
6
0.5
5
0.6
5
0.7
4
0.8
4
0.9
4
1.0
3
1.1
3
1.2
3
vary and hold ρ constant, as is pointed out in Table
5
2
6
3
7
3
8
3
9
4
10
4
11
4
12
5
13
5
14
5
15
5
16
6
These values of ns allow us to compute the effective queue-joining rate under several parameter
settings, which are at their turn used to graphically represent the gap between the individually
and socially desired optimum in observable systems.
6.3
Multi-Server Systems with Exponentially Distributed
Interarrival and Service Times (M/M/s)
6.3.1
Social Optimisation in an Observable M/M/s/n-system
In Section 3.2.2 an expression is stated from which the socially optimal threshold value in
an observable M/M/s/n-system can be derived. We make use of the numerical procedure discussed in Section 6.1.1 to get results. Note that in this case we start iterating from a value
of n = s, since lower values would lead to trivial solutions (not having to wait).
Table 6.3 summarises some outcomes for the optimal threshold ns , for different values of
ρ, while holding Rµ
= 10 and s = 3.
C
Table 6.3
Utilisation ρ
ns
0.1
28
0.2
24
0.3
21
0.4
19
75
0.5
16
0.6
14
0.7
12
0.8
10
0.9
9
1.0
8
1.1
7
1.2
7
Analogously, it is also possible to let
6.4, with ρ = 0.8, µ = 10 and s = 3.
Rµ/C
ns
Table 6.4
1
3
2
4
3
5
Rµ
C
4
6
vary and hold ρ constant, as is pointed out in Table
5
6
6
7
7
8
8
9
9
9
10
10
11
11
12
11
13
12
14
13
15
13
16
14
These values of ns allow us to compute the effective queue-joining rate under several parameter
settings, which are at their turn used to graphically represent the gap between the individually
and socially desired optimum in observable systems.
To determine the effect of the number of servers on the effective queue-joining rates, we constructed Table 6.5, where s is variable, ρ = 0.8 and Rµ
= 10.
C
Table 6.5
Servers s
ns
1
4
2
7
3
10
4
13
5
16
6
18
7
21
8
24
9
26
10
29
From this last table, we see that the formulas used in the analysis of the M/M/s-system yield
the same results for s = 1 as those used in the chapter on M/M/1. We daresay that this proves
that the formulas used here, effectively, make sense!
6.3.2
Individual Optimisation in an Unobservable M/M/s-system
In Section 3.3.1 an expression is stated from which the individually optimal threshold value
in an unobservable M/M/s-system can be derived. We make use of the numerical procedure
discussed in Section 6.1.2 to get results.
For CR = 1, µ = 10 and s = 3, we get that the critical joining/balking point is located at
λ = 28.9608 (i.e. ρ = 0.9654). For lower utilisation rates, no customer will balk. After this
point, the same effective queue-joining rate will be held, meaning that all arriving customers
will balk.
This optimal point changes as the relative benefit of the service varies. Table 6.6 demonstrates this by allowing Rµ
to change and keeping µ = 10 and s = 3.
C
Rµ/C
λ∗I
1
0.0000
2
23.6928
3
26.1592
4
27.2307
5
27.8332
6
28.2199
7
28.4893
Rµ/C
λ∗I
8
28.6878
9
28.8401
10
28.9608
11
29.0587
12
29.1397
13
29.2079
14
29.2660
Table 6.6
This data is used to construct the graphs in the main text and allows us to compare individually
and socially optimal behaviour in observable and unobservable queues.
76
Similar to the observable case, we would like to quantify the effect of the number of servers
on this unrestricted λ∗I . This study is supported by Table 6.7 where CR = 1, µ = 10 and s is
variable.
Table 6.7
Servers s
λ∗I
1
9.0000
2
18.9736
3
28.9608
4
38.9527
5
48.9469
6
58.9426
7
68.9391
It is apparent that for s = 1, the results are the same as in the chapter on M/M/1-systems.
6.3.3
Social Optimisation in an Unobservable M/M/s-system
In Section 3.3.2 an expression is stated from which the socially optimal threshold value in
an unobservable M/M/s-system can be derived. We make use of the numerical procedure discussed in Section 6.1.3 to get results.
For CR = 1, µ = 10 and s = 3, we get that the critical joining/balking point is located at
λ = 24.4602 (i.e. ρ = 0.8153). For lower utilisation rates, no customer will balk. After this
point, the same effective queue-joining rate will be held, meaning that all arriving customers
will balk.
This optimal point changes as the relative benefit of the service varies. Table 6.8 demonto change and keeping µ = 10 and s = 3.
strates this by allowing Rµ
C
Rµ/C
λ∗S
1
0.0000
2
16.6267
3
19.5027
4
21.0443
5
22.0540
6
22.7825
7
23.3405
Rµ/C
λ∗S
8
23.7859
9
24.1521
10
24.4602
11
24.7241
12
24.9535
13
25.1553
14
25.3347
Table 6.8
This data is used to construct the graphs in the main text and allows us to compare individually
and socially optimal behaviour in observable and unobservable queues.
Similar to the individual case, we would like to quantify the effect of the number of servers on
this unrestricted λ∗S . This study is supported by Table 6.9 where CR = 1, µ = 10 and s is
variable.
Table 6.9
Servers s
λ∗S
1
6.8378
2
15.4920
3
24.4602
4
33.5906
5
42.8246
6
52.1321
7
61.4954
It is apparent that for s = 1, the results are the same as in the chapter on M/M/1-systems.
77
6.4
Single-Server Systems with Hyperexponentially Distributed Interarrival Times (H2/M/1)
6.4.1
Numerical Solution for H2 /M/1/n
In Section 4.2 a set of balance equations is formulated that cannot be readily distilled into a
single expression for all steady-state probabilities. We will therefore make use of the numerical
procedure discussed in Section 6.1.4 to get to solutions for the H2 /M/1/n-system.
As an example, Table 6.10 shows the outcome of the procedure when setting n = 10, ρ = 0.8,
µ = 10 and Cτ2 = 4. The sum of all these probabilities is 1.
k
p(k, 1)
p(k, 2)
0
0.0418
0.2310
1
0.0594
0.0416
2
0.0541
0.0378
3
0.0494
0.0342
4
0.0453
0.0310
5
0.0420
0.0280
6
0.0395
0.0251
7
0.0382
0.0224
8
0.0387
0.0196
9
0.0420
0.0165
10
0.0496
0.0129
These steady-state probabilities are then used to calculate the loss probability qn and the expected number of customers in the system Ln . The individually optimal queue-joining rate λO,I
e
can be found immediately.
In order to find the social λO,S
e , we first need to apply the numerical procedure described
in Section 6.1.1.
Table 6.11 summarises some outcomes for the optimal threshold ns , for different values of
ρ, while holding Rµ
= 10 and Cτ2 = 4.
C
Table 6.11
Utilisation ρ
ns
0.1
9
0.2
7
0.3
6
0.4
5
0.5
5
0.6
4
0.7
4
0.8
4
0.9
4
1.0
4
1.1
3
1.2
3
Analogously, it is also possible to let Rµ
vary and hold ρ constant, as is pointed out in Table
C
2
6.12, with ρ = 0.8, µ = 10 and Cτ = 4.
Table 6.12
Rµ/C
ns
1
1
2
1
3
2
4
2
5
2
6
3
7
3
8
3
9
4
10
4
11
4
12
5
13
5
14
5
15
5
16
6
These values of ns allow us to compute the effective queue-joining rate under several parameter
settings, which are at their turn used to graphically represent the gap between the individually
and socially desired optimum in observable systems.
78
To determine the effect of the squared coefficient of variation on the effective queue-joining
rates, we constructed Table 6.13, where Cτ2 is variable, ρ = 0.8 and Rµ
= 10.
C
Table 6.13
SCV Cτ2
ns
1
4
2
4
3
4
4
4
5
4
6
4
7
4
8
4
9
3
10
3
From this last table, we see that the formulas used in the analysis of the H2 /M/1/n-system
yield the same results for Cτ2 = 1 as those used in the chapter on M/M/1/n.
6.4.2
Individual Optimisation in an Unobservable H2 /M/1-system
In Section 4.3.1 an expression is stated from which the individually optimal threshold value
in an unobservable H2 /M/1-system can be derived. We make use of the numerical procedure
discussed in Section 6.1.2 to get results.
For CR = 1, µ = 10 and Cτ2 = 4, we get that the critical joining/balking point is located
at λ = 7.8618. For lower utilisation rates, no customer will balk. After this point, the same
effective queue-joining rate will be held, meaning that all arriving customers will balk.
This optimal point changes as the relative benefit of the service varies. Table 6.14 demonstrates this by allowing Rµ
to change and keeping µ = 10 and Cτ2 = 4.
C
Rµ/C
λ∗I
1
0.0000
2
3.3969
3
4.8314
4
5.7095
5
6.3243
6
6.7843
7
7.1427
Rµ/C
λ∗I
8
7.4300
9
7.6654
10
7.8618
11
8.0280
12
8.1705
13
8.2940
14
8.4020
Table 6.14
This data is used to construct the graphs in the main text and allows us to compare individually
and socially optimal behaviour in observable and unobservable queues.
Similar to the observable case, we would like to quantify the effect of the squared coefficient of
variation on this unrestricted λ∗I . This study is supported by Table 6.15 where CR = 1, µ = 10
and Cτ2 is variable.
Table 6.15
SCV Cτ2
λ∗I
1
9.0000
2
8.5744
3
8.1960
4
7.8618
5
7.5676
6
7.3093
7
7.0826
It is apparent that for Cτ2 = 1, the results are the same as in the chapter on M/M/1-systems.
79
6.4.3
Social Optimisation in an Unobservable H2 /M/1-system
In Section 4.3.2 an expression is stated from which the individually optimal threshold value
in an unobservable H2 /M/1-system can be derived. We make use of the numerical procedure
discussed in Section 6.1.3 to get results.
For CR = 1, µ = 10 and Cτ2 = 4, we get that the critical joining/balking point is located
at λ = 5.3867. For lower utilisation rates, no customer will balk. After this point, the same
effective queue-joining rate will be held, meaning that all arriving customers will balk.
This optimal point changes as the relative benefit of the service varies. Table 6.16 demonstrates this by allowing Rµ
to change and keeping µ = 10 and Cτ2 = 4.
C
Rµ/C
λ∗S
1
0.0000
2
1.9351
3
2.9034
4
3.5413
5
4.0146
6
4.3896
7
4.6989
Rµ/C
λ∗S
8
4.8702
9
5.1206
10
5.3867
11
5.5632
12
5.7213
13
5.8640
14
5.9937
Table 6.16
This data is used to construct the graphs in the main text and allows us to compare individually
and socially optimal behaviour in observable and unobservable queues.
Similar to the individual case, we would like to quantify the effect of the squared coefficient of
variation on this unrestricted λ∗S . This study is supported by Table 6.17 where CR = 1, µ = 10
and Cτ2 is variable.
Table 6.17
SCV Cτ2
λ∗S
1
6.8378
2
6.2223
3
5.7522
4
5.3867
5
5.1006
6
4.8753
7
4.6962
It is apparent that for Cτ2 = 1, the results are the same as in the chapter on M/M/1-systems.
80
6.5
Single-Server Systems with Hyperexponentially Distributed Service Times (M/H2/1)
6.5.1
Numerical Solution for M/H2 /1/n
In Section 5.2 a set of balance equations is formulated that cannot be readily distilled into a
single expression for all steady-state probabilities. We will therefore make use of the numerical
procedure discussed in Section 6.1.4 to get to solutions for the M/H2 /1/n-system.
As an example, Table 6.18 shows the outcome of the procedure when setting n = 10, ρ = 0.8,
µ = 10 and Cx2 = 4. Recall that here the probability of having no customers in the system
is captured by a single variable p(0). The p(k, j)’s are therefore only defined for k ≥ 1. As
always, the sum of all these probabilities is 1.
k
p(k, 1)
p(k, 2)
1
0.1174
0.0374
2
0.0649
0.0383
3
0.0419
0.0368
4
0.0311
0.0345
5
0.0255
0.0319
6
0.0221
0.0295
7
0.0198
0.0271
8
0.0180
0.0249
9
0.0164
0.0229
10
0.0074
0.0813
The probability of an empty system is given by p(0) = 0.2709. These steady-state probabilities
are then used to calculate the loss probability qn and the expected number of customers in the
can be found immediately.
system Ln . The individually optimal queue-joining rate λO,I
e
In order to find the social λO,S
e , we first need to apply the numerical procedure described
in Section 6.1.1.
Table 6.19 summarises some outcomes for the optimal threshold ns , for different values of
= 10 and Cx2 = 4.
ρ, while holding Rµ
C
Table 6.19
Utilisation ρ
ns
0.1
6
0.2
5
0.3
4
0.4
4
0.5
4
0.6
4
0.7
3
0.8
3
0.9
3
1.0
3
1.1
3
1.2
3
Analogously, it is also possible to let Rµ
vary and hold ρ constant, as is pointed out in Table
C
2
6.20, with ρ = 0.8, µ = 10 and Cx = 4.
Table 6.20
Rµ/C
ns
1
1
2
1
3
2
4
2
5
2
6
2
7
2
8
3
9
3
10
3
11
4
12
4
13
4
14
4
15
5
16
5
These values of ns allow us to compute the effective queue-joining rate under several parameter
settings, which are at their turn used to graphically represent the gap between the individually
and socially desired optimum in observable systems.
81
To determine the effect of the squared coefficient of variation on the effective queue-joining
rates, we constructed Table 6.21, where Cx2 is variable, ρ = 0.8 and Rµ
= 10.
C
Table 6.21
SCV Cx2
ns
1
4
2
4
3
4
4
3
5
3
6
3
7
3
8
3
9
3
10
3
From this last table, we see that the formulas used in the analysis of the M/H2 /1/n-system
yield the same results for Cx2 = 1 as those used in the chapter on M/M/1/n.
6.5.2
Individual Optimisation in an Unobservable M/H2 /1-system
In Section 5.3.1 an expression is stated from which the individually optimal threshold value
in an unobservable M/H2 /1-system can be derived. We make use of the numerical procedure
discussed in Section 6.1.2 to get results.
For CR = 1, µ = 10 and Cx2 = 4, we get that the critical joining/balking point is located
at λ = 7.8259. For lower utilisation rates, no customer will balk. After this point, the same
effective queue-joining rate will be held, meaning that all arriving customers will balk.
This optimal point changes as the relative benefit of the service varies. Table 6.22 demonstrates this by allowing Rµ
to change and keeping µ = 10 and Cx2 = 4.
C
Rµ/C
λ∗I
1
0.0000
2
2.8568
3
4.4441
4
5.4543
5
6.1536
6
6.6664
7
7.0586
Rµ/C
λ∗I
8
7.3682
9
7.6189
10
7.8259
11
7.9999
12
8.1480
13
8.2758
14
8.3870
Table 6.22
This data is used to construct the graphs in the main text and allows us to compare individually
and socially optimal behaviour in observable and unobservable queues.
Similar to the observable case, we would like to quantify the effect of the squared coefficient of
variation on this unrestricted λ∗I . This study is supported by Table 6.23 where CR = 1, µ = 10
and Cx2 is variable.
Table 6.23
SCV Cx2
λ∗I
1
9.0000
2
8.5713
3
8.1817
4
7.8259
5
7.4998
6
7.1998
7
6.9229
It is apparent that for Cx2 = 1, the results are the same as in the chapter on M/M/1-systems.
82
6.5.3
Social Optimisation in an Unobservable M/H2 /1-system
In Section 5.3.2 an expression is stated from which the individually optimal threshold value
in an unobservable M/H2 /1-system can be derived. We make use of the numerical procedure
discussed in Section 6.1.3 to get results.
For CR = 1, µ = 10 and Cx2 = 4, we get that the critical joining/balking point is located
at λ = 5.3376. For lower utilisation rates, no customer will balk. After this point, the same
effective queue-joining rate will be held, meaning that all arriving customers will balk.
This optimal point changes as the relative benefit of the service varies. Table 6.24 demonstrates this by allowing Rµ
to change and keeping µ = 10 and Cx2 = 4.
C
Rµ/C
λ∗S
1
0.0000
2
1.5486
3
2.5465
4
3.2581
5
3.7984
6
4.2266
7
4.5768
Rµ/C
λ∗S
8
4.8702
9
5.1206
10
5.3376
11
5.5280
12
5.6968
13
5.8478
14
5.9840
Table 6.24
This data is used to construct the graphs in the main text and allows us to compare individually
and socially optimal behaviour in observable and unobservable queues.
Similar to the individual case, we would like to quantify the effect of the squared coefficient of
variation on this unrestricted λ∗S . This study is supported by Table 6.25 where CR = 1, µ = 10
and Cx2 is variable.
Table 6.25
SCV Cx2
λ∗S
1
6.8378
2
6.2205
3
5.7361
4
5.3376
5
5.0001
6
4.7086
7
4.4531
It is apparent that for Cx2 = 1, the results are the same as in the chapter on M/M/1-systems.
83
Chapter 7
Conclusion
Now that the analysis is behind us, it is time to take back to the very first section of this thesis.
As a summary, we will concisely bundle all the results to answer the three research questions
that were set up.
7.1
Individual vs. Social Behaviour
The first research question considers the difference between individually and socially optimal
behaviour. We found that the socially optimal effective queue-joining rate is always lower than
or equal to the individual one. Selfishly acting customers thus decrease social welfare through
their queueing behaviour, resulting in longer queues and longer queueing times.
This finding is true for both observable and unobservable systems. It has been tested in
single-server and multi-server environments, as well as in situations with increased variability
in the interarrival or service times.
To make customers act in the socially desired way, we have access to two correcting instruments.
An appropriately sized admission fee or waiting toll sets the balking threshold (observable) or
balking probability (unobservable) to the socially preferred value. Since admission fees imply
lower transfer payments and are less volatile, it is suggested that service providers implement
these instead of a toll on waiting.
84
7.2
Observable vs. Unobservable Systems
As we are moving on to the second research question, we are effectively getting to the core of
this thesis. Our main goal is namely to determine whether or not it is beneficial for the service
provider to reveal information on the queue length to customers. Whereas the previous question has been investigated on for decades, the subject of this section has only gained interest
of the academic world in recent years. It is here that this thesis creates its added value.
Our analysis has shown that the decision to make an unobservable queue observable is not
straightforward. The optimal tactic depends on the specific values of the system’s parameters.
We can, however, formulate three general principles that a service provider should keep in mind.
1. The range in which equality can occur, in terms of the relative value of the service, is
bigger under socially optimal customer behaviour.
2. Equality under individually optimal customer behaviour occurs for lower CR , i.e. for systems with a lower service gain, when compared to equivalent socially optimal behaviour.
3. In general, the incentive to reveal the queue length increases as the relative service gain
R
is low and the utilisation rate ρ is high.
C
These guidelines remain valid when applied in multi-server situations and cases where additional
variation is present.
7.3
Effect of Additional Variation
The final research question is aimed at expanding the scope of this thesis to more complicated
(i.e. closer to real-life) systems. Modeling arrival and service rates using an exponential distribution can underestimate the true variability that a system is faced with. We have therefore
considered the option of using hyperexponential distributions that allow us to account for more
variation.
It became immediately clear that this additional variation has a detrimental effect on the
effective queue-joining rates, under both types of behaviour and both types of system. In addition, the social welfare is highly diminished. A service provider should thus always strive to
minimise the degree of variability that his system has to cope with.
85
Regarding the question whether or not to reveal information on the queue length to customers,
we can construct three more principles for systems with increased variation.
4.
For systems that have to cope with higher variation in the interarrival or service times,
it is beneficial to start revealing the queue length at lower utilisation rates ρ.
5.
For systems that have to cope with higher variation in the interarrival or service times,
the incentive to reveal the queue length is present for higher relative service gains CR .
6.
Systems with higher variation in their interarrival or service times experience a bigger gap
between the optimal effective queue-joining rates in an observable versus an unobservable
system. The decision whether or not to reveal the queue length will thus have a bigger
impact on the service provider’s profit.
In short, we can state that the additional variation makes revealing information on the queue
length both more attractive and necessary!
Our last remark constitutes the difference between variation in the arrival or service rates.
Their impact on welfare is much the same, though some small dissimilarities can be observed.
In observable systems, the negative effect of variation is slightly smaller when it occurs in the
service times as opposed to the interarrival times. The opposite is true for unobservable systems.
Nonetheless, this difference pales in comparison to its overall detrimental effect. Reducing
variation should thus always be priority number one!
7.4
Further Research
I would like to devote the final section of this thesis to suggest some possible directions for
further research. Of course, these suggestions are based on what comes to mind without having
truly investigated on them. Nevertheless, they provide some interesting food for thought.
A first option that I want to discuss is the one of having customers with a different perception
of the value of the service. For example, a customer ringing up the call center of his cable
provider because he has a question about his invoice (”Why do I have to pay five euros more
than last month?”) will perceive the service to be rather invaluable. The probability that this
customer balks will thus be relatively high. Opposed to this are customers that call because
they do not receive their television/mobile/Internet-signal anymore. They value the service of
the call center a lot higher and are willing to wait much longer.
86
The concept of having two types of customers is not new to the field of queueing theory.
Hassin and Haviv [3] and Larsen [5] have already provided a start to think in this direction
regarding individually and socially optimal behaviour. The comparison between observable and
unobservable systems and the question whether or not to reveal the queue length remain open,
however.
In a sense, one could argue that this is a form of increased variability in the arrival process (if
we consider the expected service time and its distribution equal for both types). A different
valuation of the service implies after all a different strategy in the joining/balking decision by
customers. We would thus expect similar results as in Chapter 4.
Another option is to assume increasing marginal costs as the time in queue increases. We
suggest to model the costs as C.Ts (λ)α , where α ≥ 1. Throughout this thesis we have considered these costs to be linear (i.e α = 1), while this proposed model most likely lies closer to the
real-life situation. Several psychological effects come into play, making this a well supported
assumption [11]. After a certain amount of time, the sense of waste and uncertainty start to
build up stress and anxiety in an individual customer. These factors increase his cost of waiting
as the customer progresses in the queue.
In a sense, the list with assumptions stated in the beginning of Chapter 2 provides many
possibilities for more profound exploration of the matter. Implying other queueing disciplines
or allowing for reneging for example are also interesting opportunities to investigate on. However, I personally consider the two options discussed here to be the most promising and alluring
directions for further research.
87
88
List of Figures
Chapter 1
Figure 1.1
Notation for Effective Queue-joining rates
3
Chapter 2
Figure 2.1
Figure
Figure
Figure
Figure
2.2
2.3
2.4
2.5
Figure 2.6
Figure 2.7
Figure 2.8
Figure 2.9
Figure 2.10
Figure
Figure
Figure
Figure
2.11
2.12
2.13
2.14
Evolution of net social gains in an Observable M/M/1-system, R = C = 10, λ = 8,
µ = 10
R
= 1, µ = 10
Optimal Threshold values in an M/M/1-system, C
R
Individual vs. Social Optimisation in an Observable M/M/1-system, C
= 1, µ = 10
Individual vs. Social Optimisation in an Observable M/M/1-system, ρ = 0.8, µ = 10
Size of the Admission Fee/Waiting Toll in an Observable M/M/1-system, C = 10,
µ = 10
R
= 1, µ = 10
Individual vs. Social Optimisation in an Unobservable M/M/1-system, C
Individual vs. Social Optimisation in an Unobservable M/M/1-system, ρ = 0.8, µ = 10
Size of the Admission Fee/Waiting Toll in an Unobservable M/M/1-system, C = 10,
µ = 10
R
= 1,
Individual Optimisation in Observable vs. Unobservable M/M/1-systems, C
µ = 10
Individual Optimisation in Observable vs. Unobservable M/M/1-systems, ρ = 0.8,
µ = 10
R
Social Optimisation in Observable vs. Unobservable M/M/1-systems, C
= 1, µ = 10
Social Optimisation in Observable vs. Unobservable M/M/1-systems, ρ = 0.8, µ = 10
Equality of Individual rates in Observable vs. Unobservable M/M/1-systems
Equality of Social rates in Observable vs. Unobservable M/M/1-systems
10
11
12
12
13
15
15
17
18
18
20
20
21
21
Chapter 3
Figure 3.1
Figure
Figure
Figure
Figure
3.2
3.3
3.4
3.5
Figure 3.6
Figure 3.7
Figure 3.8
Evolution of net social gains in an Observable M/M/3-system, R = C = 10, λ = 8,
µ = 10
R
= 1, µ = 10
Optimal Threshold values in an M/M/3-system, C
R
Individual vs. Social Optimisation in an Observable M/M/3-system, C
= 1, µ = 10
Individual vs. Social Optimisation in an Observable M/M/3-system, ρ = 0.8, µ = 10
Size of the Admission Fee/Waiting Toll in an Observable M/M/3-system, C = 10,
µ = 10
R
Individual vs. Social Optimisation in an Unobservable M/M/3-system, C
= 1, µ = 10
Individual vs. Social Optimisation in an Unobservable M/M/3-system, ρ = 0.8, µ = 10
Size of the Admission Fee/Waiting Toll in an Unobservable M/M/3-system, C = 10,
µ = 10
XI
25
25
26
26
27
29
29
30
Figure 3.9
Figure 3.10
Figure
Figure
Figure
Figure
3.11
3.12
3.13
3.14
R
Individual Optimisation in Observable vs. Unobservable M/M/3-systems, C
= 1,
µ = 10
Individual Optimisation in Observable vs. Unobservable M/M/3-systems, ρ = 0.8,
µ = 10
R
Social Optimisation in Observable vs. Unobservable M/M/3-systems, C
= 1, µ = 10
Social Optimisation in Observable vs. Unobservable M/M/3-systems, ρ = 0.8, µ = 10
R
= 1, ρ = 0.8, µ = 10
Effect of the number of servers on effective queue-joining rates, C
R
Loss probabilities under Individual Optimisation, C = 1, µ = 10
31
32
32
33
34
35
Chapter 4
Figure 4.1
Figure 4.2
Figure 4.3
Figure 4.4
Figure 4.5
Figure 4.6
Figure 4.7
Figure 4.8
Figure 4.9
Figure 4.10
Figure 4.11
Figure 4.12
Figure 4.13
Figure 4.14
Figure 4.15
Figure 4.16
Figure 4.17
Figure 4.18
Conceptual Drawing of Hyperexponential interarrival times
Relation between the SCV and the probability of joining the first phase
Evolution of net social gains in an Observable H2 /M/1-system, R = C = 10, λ = 8,
µ = 10, Cτ2 = 4
R
= 1, µ = 10, Cτ2 = 4
Optimal Threshold values in an H2 /M/1-system, C
R
Individual vs. Social Optimisation in an Observable H2 /M/1-system, C
= 1, µ = 10,
2
Cτ = 4
Individual vs. Social Optimisation in an Observable H2 /M/1-system, ρ = 0.8, µ = 10,
Cτ2 = 4
Size of the Admission Fee/Waiting Toll in an Observable H2 /M/1-system, C = 10,
µ = 10, Cτ2 = 4
R
Individual vs. Social Optimisation in an Unobservable H2 /M/1-system, C
= 1, µ = 10,
Cτ2 = 4
Individual vs. Social Optimisation in an Unobservable H2 /M/1-system, ρ = 0.8,
µ = 10, Cτ2 = 4
Size of the Admission Fee/Waiting Toll in an Unobservable H2 /M/1-system, C = 10,
µ = 10, Cτ2 = 4
R
Individual Optimisation in Observable vs. Unobservable H2 /M/1-systems, C
= 1,
2
µ = 10, Cτ = 4
Individual Optimisation in Observable vs. Unobservable H2/M/1-systems, ρ = 0.8,
µ = 10, Cτ2 = 4
R
Social Optimisation in Observable vs. Unobservable H2 /M/1-systems, C
= 1, µ = 10,
2
Cτ = 4
Social Optimisation in Observable vs. Unobservable H2 /M/1-systems, ρ = 0.8, µ = 10,
Cτ2 = 4
R
Effect of Variation in the Interarrival Times in an Observable System, C
= 1, ρ = 0.8,
µ = 10
R
Effect of Variation in the Interarrival Times in an Unobservable System, C
= 1,
ρ = 0.8, µ = 10
R
Effect of Variation in the Interarrival Times under Individual Behaviour, C
= 1,
ρ = 0.8, µ = 10
R
Effect of Variation in the Interarrival Times under Individual Behaviour, C
= 1,
ρ = 0.8, µ = 10
XII
36
38
43
43
44
44
45
47
47
48
49
50
51
51
53
53
54
54
Chapter 5
Figure 5.1
Figure 5.2
Figure 5.3
Figure 5.4
Figure 5.5
Figure 5.6
Figure 5.7
Figure 5.8
Figure 5.9
Figure 5.10
Figure 5.11
Figure 5.12
Figure 5.13
Figure 5.14
Figure 5.15
Figure 5.16
Evolution of net social gains in an Observable M/H2 /1-system, R = C = 10, λ = 8,
µ = 10, Cτ2 = 4
R
Optimal Threshold values in an M/H2 /1-system, C
= 1, µ = 10, Cτ2 = 4
R
Individual vs. Social Optimisation in an Observable M/H2 /1-system, C
= 1, µ = 10,
2
Cτ = 4
Individual vs. Social Optimisation in an Observable M/H2 /1-system, ρ = 0.8, µ = 10,
Cτ2 = 4
Size of the Admission Fee/Waiting Toll in an Observable M/H2 /1-system, C = 10,
µ = 10, Cτ2 = 4
R
Individual vs. Social Optimisation in an Unobservable M/H2 /1-system, C
= 1, µ = 10,
2
Cτ = 4
Individual vs. Social Optimisation in an Unobservable M/H2 /1-system, ρ = 0.8,
µ = 10, Cτ2 = 4
Size of the Admission Fee/Waiting Toll in an Unobservable M/H2 /1-system, C = 10,
µ = 10, Cτ2 = 4
R
Individual Optimisation in Observable vs. Unobservable M/H2 /1-systems, C
= 1,
2
µ = 10, Cτ = 4
Individual Optimisation in Observable vs. Unobservable M/H2/1-systems, ρ = 0.8,
µ = 10, Cτ2 = 4
R
= 1, µ = 10,
Social Optimisation in Observable vs. Unobservable M/H2 /1-systems, C
2
Cτ = 4
Social Optimisation in Observable vs. Unobservable M/H2 /1-systems, ρ = 0.8, µ = 10,
Cτ2 = 4
R
Effect of Variation in the Service Times in an Observable System, C
= 1, ρ = 0.8,
µ = 10
R
= 1,
Effect of Variation in the Service Times in an Unobservable System, C
ρ = 0.8, µ = 10
R
Effect of Variation in the Service Times under Individual Behaviour, C
= 1,
ρ = 0.8, µ = 10
R
= 1,
Effect of Variation in the Service Times under Social Behaviour, C
ρ = 0.8, µ = 10
XIII
60
61
61
62
62
64
65
65
66
67
67
68
69
70
70
71
XIV
List of Tables
Chapter 1
Table 1.1
Table 1.2
Table 1.3
General System Parameters
Parameters used in the analysis of Observable Systems
Parameters used in the analysis of Unobservable Systems
3
4
6
Chapter 4
Table 4.1
Quantities used in the analysis of the H2 /M/1-and H2 /M/1/n-system
39
Quantities used in the analysis of the M/H2 /1-and M/H2 /1/n-system
56
Socially Optimal Threshold values in an Observable M/M/1-system, Rµ
C = 10
Socially Optimal Threshold values in an Observable M/M/1-system, ρ = 0.8, µ = 10
Socially Optimal Threshold values in an Observable M/M/3-system, Rµ
C = 10
Socially Optimal Threshold values in an Observable M/M/3-system, ρ = 0.8, µ = 10
Socially Optimal Threshold values in an Observable M/M/3-system, Rµ
C = 10, ρ = 0.8
Unrestricted effective arrival rates under Individual Optimisation in an Unobservable
M/M/3-system, µ = 10
Unrestricted effective arrival rates under Individual Optimisation in an Unobservable
R
M/M/3-system, C
= 1, µ = 10
Unrestricted effective arrival rates under Social Optimisation in an Unobservable
M/M/3-system, µ = 10
Unrestricted effective arrival rates under Social Optimisation in an Unobservable
R
M/M/3-system, C
= 1, µ = 10
Steady-state probabilities in an H2 /M/1-system, n = 10, ρ = 0.8, µ = 10, Cτ2 = 4
2
Socially Optimal Threshold values in an Observable H2 /M/1-system, Rµ
C = 10, Cτ = 4
Socially Optimal Threshold values in an Observable H2 /M/1-system, ρ = 0.8, µ = 10,
Cτ2 = 4
Socially Optimal Threshold values in an Observable H2 /M/1-system, Rµ
C = 10, ρ = 0.8
Unrestricted effective arrival rates under Individual Optimisation in an Unobservable
H2 /M/1-system, µ = 10, Cτ2 = 4
Unrestricted effective arrival rates under Individual Optimisation in an Unobservable
R
H2 /M/1-system, C
= 1, µ = 10
Unrestricted effective arrival rates under Social Optimisation in an Unobservable
H2 /M/1-system, µ = 10, Cτ2 = 4
75
75
75
76
76
76
Chapter 5
Table 5.1
Chapter 6
Table
Table
Table
Table
Table
Table
6.1
6.2
6.3
6.4
6.5
6.6
Table 6.7
Table 6.8
Table 6.9
Table 6.10
Table 6.11
Table 6.12
Table 6.13
Table 6.14
Table 6.15
Table 6.16
XV
77
77
77
78
78
78
79
79
79
80
Table 6.17
Table 6.18
Table 6.19
Table 6.20
Table 6.21
Table 6.22
Table 6.23
Table 6.24
Table 6.25
Unrestricted effective arrival rates under Social Optimisation in an Unobservable
R
H2 /M/1-system, C
= 1, µ = 10
Steady-state probabilities in an M/H2 /1-system, n = 10, ρ = 0.8, µ = 10, Cτ2 = 4
2
Socially Optimal Threshold values in an Observable H2 /M/1-system, Rµ
C = 10, Cτ = 4
Socially Optimal Threshold values in an Observable H2 /M/1-system, ρ = 0.8, µ = 10,
Cτ2 = 4
Socially Optimal Threshold values in an Observable H2 /M/1-system, Rµ
C = 10, ρ = 0.8
Unrestricted effective arrival rates under Individual Optimisation in an Unobservable
M/H2 /1-system, µ = 10, Cτ2 = 4
Unrestricted effective arrival rates under Individual Optimisation in an Unobservable
R
M/H2 /1-system, C
= 1, µ = 10
Unrestricted effective arrival rates under Social Optimisation in an Unobservable
M/H2 /1-system, µ = 10, Cτ2 = 4
Unrestricted effective arrival rates under Social Optimisation in an Unobservable
R
= 1, µ = 10
M/H2 /1-system, C
XVI
80
81
81
81
82
82
82
83
83
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