Queueing for Timely Service:
Equilibrium Analysis and Social Efficiency∗
Rahul Jain†
Sandeep Juneja‡
Nahum Shimkin§
Abstract
We introduce the concert or the cafeteria queueing problem: A finite but large
number of customers arrive into a queue that starts serving at a specified opening
time. Customers are free to choose their arrival time (possibly before opening time),
with the purpose of finishing early while minimizing their waiting time in the queue.
This goal is captured by a cost function which is additive and linear in the waiting time
and service completion time, with coefficients that may be class dependent. Besides
the illustrative examples of the concert or cafeteria queues, the proposed model may
also explain queues in many similar settings such as ticket purchase, DMV offices at
opening time, movie theaters, communications and computer networks, and at retail
stores at opening time during peak shopping season. We analyze the system in the
asymptotic regime and motivate a fluid model for the resultant queueing system. In
the fluid setting, we characterize the Nash equilibrium points, show that they induce
a unique arrival profile for each customer class, and explicitly derive their structure.
In a single class setting, we show that the price of anarchy (the efficiency loss relative
to the socially optimal solution) equals two, while in multi-class settings we develop
tight upper and lower bounds for the price of anarchy. In addition, in the simple single
class setting we discuss and quantify certain management means through which the
price of anarchy may be reduced.
∗
INFORMS MSOM 2010 Conference – Service Management SIG, Haifa, June 2010.
EE & ISE departments, Viterbi School of Engineering, University of Southern California, Los Angeles,
CA. email: [email protected]
‡
School of Technology and Computer Science, Tata Institute of Fundamental Research, Mumbai, India.
email: [email protected]
§
Department of Electrical Engineering, Technion – Israel Institute of Technology, Haifa 32000, Israel.
e-mail: [email protected]
†
1
Key words: Fluid model, finite-duration queues, Nash equilibrium, arrival profile, Price of
Anarchy reduction.
1
Introduction
In this paper, we introduce the concert queueing game. This is motivated by the following
scenario. Before going to say, a rock concert in a theater with unassigned seats where a
large queue may be anticipated, one faces the following dilemma: Should one go late when
queues are smaller but the best seats may already be taken, or go early to get the best
seats but one may have to wait in a long queue? Similarly, in cafeterias that open at a
specific time for lunch, should one go early when one is really hungry but the queues may
be longer, or go hungry longer to avoid the long queues? Similar trade-offs govern customer
decision making in many queueing situations such as visiting a retail store on the day of a
huge sale, visit to the DMV office, to a movie theater, etc. In communication networks, for
example, congestion is typically not uniform throughout the day. There is greater traffic
during working hours than at night. Thus, if one wants to download a large file, one has the
option of either downloading it during the day but experience a longer download time or
at a less convenient time at night when the download speeds would be considerably faster.
In this paper, we propose a modeling approach that can be used to study system behaviors
when strategic players are faced by such utility-queueing delay trade-offs.
In our model we assume that there are a large but fixed number of customers that need to
be served by a server in a first come first serve manner. The server at the queue becomes
active at a particular time. We allow the flexibility that the customers can choose to arrive
and queue up both before and after that time. The cost structure of each customer is
additive and linear in the waiting time and in time at which service is received. Multiple
classes of customers are allowed in that the cost coefficients can take different values. We
primarily focus on the case where the number of distinct classes is finite, however we do
briefly discuss the case where we have a continuum of classes.
Under the assumption that each customer implements an independent and possibly mixed
strategy, i.e., selects her time of arrival as a sample from a probability distribution, we
develop a fluid limit for the resultant queueing system. The resulting fluid model offers
a great deal of analytical simplification. In particular, we show that in the fluid system
the arrival distribution can be modeled as a non-atomic game [15] for which we find a
2
unique Nash equilibrium arrival profile. A key simplification is achieved by noting that
in equilibrium the queue never becomes empty until every customer has been served. We
also identify the cost of the socially optimal solution associated with this game. Price of
anarchy (PoA) is a quantitative measure that has been of interest recently to capture the
relative efficiency loss of the equilibrium solution. It equals the ratio of the social cost of
the equilibrium solution to that of the optimal one. In the single class setting, we note that
PoA of this game is 2 for all parameter values. In the multi-class setting, we develop tight
upper bounds and corresponding lower bounds on PoA that depend only on the range of
the cost parameters across customer classes.
We also discuss a few ways to limit this PoA in the simpler single class setting. One
involves allowing certain proportion of the population to be served only after a specified time
after the service facility opens. The second involves dividing the population in segments
and assigning different priorities to each segment. The third involves charging a tariff to
customers who get served before a specified time. We note that by sufficiently segmenting
the population optimally along any of these three ways, PoA can be brought arbitrarily
close to one and we identify the associated rate of convergence.
In many settings, costs for a given customer may be modeled as a sum of a term proportional
to the delay in the queue and another term proportional to the number of customers that
have been served before this customer. The latter differs from our earlier assumption of
cost depending on the time at which service is received. This, for instance, is relevant in
concert hall, movie theater and similar settings where the quality of available seats depends
on how many people have entered (or purchased tickets) before. We will show in Section
3.2 that our fluid model captures this scenario as well.
Strategic models involving queues with self-optimizing customers have been extensively
studied in the last four decades, spanning problems of admission control, routing, reneging,
choice of priorities, pricing, and related issues. A sizable part of this literature is summarized in the monograph [5]. A central issue in this context is the comparison of the
individual equilibrium and the socially optimal solution. This may be traced back to the
seminal paper [11] of Naor, who considered these two solutions in the context of admission control to a single-server queue with full state information, and suggested pricing as a
means to induce the social optimum. For example, [9] provides bounds on the PoA for the
problem of routing into n parallel servers, and [3] studies the PoA in Naor’s model.
Equilibrium arrival patterns to queues with finite service period were apparently first considered in [4], where a Poisson-distributed number of homogeneous customers may choose
3
their arrival times with the goal of minimizing their waiting time. In this model, customers
are indifferent to the time-of-day in which service is completed. Several extensions and
variations of this model have been considered, e.g., in [13, 7, 6], and are further described
in [5](Chapter 6) and in [6]. The model of [18] incorporates preferences for early service,
but does so within a multiple shift scheme where the service period is divided into evenlyspaced shifts, and the waiting time in each shift is determined by the number of customers
who choose this shift.
In the transportation literature, equilibrium trip-timing patterns were extensively studied in
the context of the so-called bottleneck or morning commute problem. [17] introduced a fluid
flow model, where homogeneous commuters choose their departure time for travel through
a single bottleneck of fixed flow capacity. The cost function for each commuter includes a
penalty for arriving early or late to the destination (relative to the desired arrival time), in
addition to the cost of delay in the bottleneck. Pointers to the extensive ensuing literature
regarding this model and its generalizations may be found in [10]. In particular, [12]
introduced commuter heterogeneity in terms of their (linear) cost coefficients, in addition
to the required arrival time. [10] provides existence and uniqueness results for the multiclass
model with nonlinear costs, under fairly general conditions. We note that our fluid model
can be considered a special case of these models with the desired arrival times all set to
zero though the bottleneck model does not have a predetermined opening time. However,
the explicit expressions we present here for the equilibrium and the analysis of the PoA are
new. In Section 6 we present a number of ways for limiting price of anarchy that involve
simultaneous existence of separate queues of customers. This analysis is new and perhaps
not very relevant to the transportation single bottleneck setting.
The organization of this paper is as follows: In Section 2 we spell out the mathematical
framework for the queueing system that we consider. Here we conduct asymptotic analysis
and derive the fluid limit of this queueing system. In Section 3, we show the existence of
a unique Nash equilibrium arrival profile in the single class setting. This restriction helps
us illustrate the key ideas without the notational burden. We generalize the results to
the multi-class settings in Section 4 where we consider finite number of classes. In both
Sections 3 and 4, we also develop price of anarchy related computations. In Section 5 we
discuss several ways to limit price of anarchy. Finally, we end with a brief conclusion in
Section 6.
4
2
Mathematical Framework and The Fluid Limit
We briefly consider here a fluid scaling that motivates the fluid model considered in the rest
of the paper. Full treatment of the stochastic system and its convergence to the fluid model
does not fit into the present paper and will be given elsewhere. Consider a series of queueing
systems indexed by n, the relative population size, that we analyze as n → ∞. For the
system n, assume that there are Λn customers, for Λ > 0 (when Λn is not an integer, we
refer to the closest integer number). Each customer i independently picks an arrival time
as a sample from a probability distribution with cumulative distribution function1 (CDF)
Fni (·). The queue begins to serve at time 0. The service times of the customers form an
i.i.d. sequence (Vi : 1 ≤ i ≤ Λn) with rate µ = 1/E(Vi ).
We first develop a fluid limit for this system. Some additional assumptions and notation
P
i
are needed for this. Let F̄n (t) = n1 Λn
i=1 Fn (t). We assume that F̄n (nt) → F (t) as n → ∞,
uniformly on compact sets (u.o.c.), where F (·) denotes the associated fluid arrival profile
that represents a positive measure on the real line of total mass Λ. Thus, F is nondecreasing, right-continuous, with F (−∞) = 0 and F (∞) = Λ.
To illustrate, consider the following simple examples where we have taken Λ = 1.
a. Let each Fni correspond to a uniform distribution on [−nT, nT ] for some T > 0,
namely Fni (t) = t+nT
on [−nT, nT ]. Then, F is a uniform distribution on [−T, T ],
2nT
t+T
that is, F (t) = 2T on this interval.
b. Let Fni correspond to the deterministic arrival time ti = (2i − n)T . Equivalently,
Fni (t) = 1{t ≥ (2i − n)T }, where 1{·} denotes an indicator function. Then F again
is the same uniform measure as before.
Let An (t) denote the number of arrivals by time t in system n. Similarly, for t ≥ 0, let
Sn (t) denote the number of service completions if the server is busy for t time units, i.e.,
m
X
Sn (t) = sup{m ≥ 0 :
Vi ≤ t}
i=1
(for t < 0 set Sn (t) = 0). Then, the queue length process is given by Qn (t) = An (t) −
Sn (Bn (t)), where Bn (t) denotes the time that the queue has been busy up to time t, namely
Bn (t) = 0 for t < 0 (as service starts at time 0), and
Z t
Bn (t) =
1{Qn (s) > 0}ds, for t ≥ 0
0
1
Throughout the paper, we identify a probability distribution on the real line with its CDF.
5
Denote s(t) = µt1{t ≥ 0}. Then Qn (t) = Xn (t) + Yn (t), where
Xn (t) = (nF̄n (t) − s(t)) + (An (t) − nF̄n (t)) − (Sn (Bn (t)) − µBn (t))
and Yn is the ‘regulator process’ Yn (t) = s(t) − µBn (t).
It then follows (see, for example, [1], Chapter 6.3) that Yn (t) = sup0≤s≤t [−Xn (s)]+ , and
Qn (t) = Xn (t) + sup [−Xn (s)]+ .
0≤s≤t
P
Let V (m) = m
i=1 Vi . Then the workload at the queue at time t equals Zn (t) = V (An (t)) −
Bn (t). Note that for t < 0 we have B(t) = 0 and s(t) = 0, which implies that Qn (t) =
Xn (t) = An (t), Yn (t) = 0, and Zn (t) = V (An (t)).
We define the normalized values of arrival, service and related process as follows: Ān (t) =
An (nt)
, Z̄n (t) = Znn(nt) , Q̄n (t) = Qnn(nt) , S̄n (t) = Snn(nt) , and B̄n (t) = Bnn(nt) . Then, as n → ∞,
n
Ān (t) → F (t) and S̄n (t) → s(t), uniformly on compact sets (u.o.c.). It then follows that
the processes (Q̄n , B̄n , Z̄n ) → (Q̄, B̄, Z̄) converge (u.o.c.), where Q̄(t) = X̄(t) = F (t) for
t < 0, and for t > 0,
Q̄(t) = X̄(t) + Ȳ (t)
(1)
where X̄(t) = F (t) − µt, Ȳ (t) = sup0≤s≤t [−X̄(s)]+ , Z̄(t) = Q̄(t)/µ, and B̄(t) = t − Ȳ (t)/µ.
The proof of these results is standard; see, for instance, Theorem 6.5 in [1] and its proof.
It follows that for large n, the scaled queueing process closely follows the fluid model with
arrival profile F (t) and constant service rate µ.
3
Game of Arrivals: Single Customer Class
We consider the concert arrival game in the fluid model setting developed in Section 2. In
our model each customer corresponds to single point in an interval [0, Λ]. These continuum
of infinitesimal customers arrive at a service facility with potential service rate µ that
becomes active at time t = 0. Customers join a single queue and are served in order of
arrival. If several customers arrive simultaneously then their order is determined randomly
and with equal probabilities. All customers could be served within Tf = Λ/µ time units.
To begin with we assume that all customers belong to a single class in that they have an
identical cost function specified as
C(w, tc ) = αw + βtc
6
where w is the customer’s waiting time in the queue, tc ≥ 0 her service completion time,
and α > 0, β > 0 are the cost sensitivities to the waiting time and service completion time.
The waiting time of a customer who arrives at time t and is placed at the end of a queue
of size q will be w = q/µ + max{0, −t} so that she completes her service and leaves the
system at tc = t + w = q/µ + max{0, t}.
Recall that F denotes the arrival profile in this fluid model so that F (−∞) = 0, F (∞) = Λ
and F (t) is right-continuous and non-decreasing in t. The following comments concern the
profile F .
Remark 1 To avoid some mathematical subtleties, we assume at the outset that the
arrival profile represented by F has no singular continuous component, and is therefore the
sum of an absolutely continuous component and a discrete component (cf. [14]).
Remark 2 As follows from the previous section, an arrival profile should be interpreted
as a deterministic summary of the arrival decisions of the individual customers, which may
themselves be deterministic or stochastic. In particular, let customer s determine her arrival
R
time according to a probability distribution with CDF Fs (t). Then F (t) = s Fs (t)U (ds),
where U is a uniform measure of total mass Λ. For example, it may be that all customers
use the same CDF Fs ≡ F/Λ (symmetric case). Alternatively, individual customers may
choose their arrival times deterministically, so that F/Λ is the fraction of customers who
choose to arrive at or before t. In any case, the resulting arrival profile F is a deterministic
function by virtue of the law of large numbers.
Given an arrival profile F , the queue process Q(t) is uniquely defined by equation (1) as
argued in the previous section. Therefore, the expected waiting time W (t) of a potential
arrival at time t is well defined. Specifically, if Q(t) is continuous at t, then the waiting
time is deterministic and given by W (t) = Q(t)/µ + max{0, −t}. If Q(t) has a jump at t
(due to an upward jump in the arrival profile F ), then the position of an arriving customer
would be uniformly distributed in [Q(t−), Q(t+)] with average Q̄(t) = 12 (Q(t−), Q(t+)),
and the expected waiting time is therefore W (t) = Q̄(t)/µ + max{0, −t}. Let WF (t) denote
the expected waiting time that corresponds to an arrival profile F .
The expected cost of a customer who arrives at t is now given by
CF (t) = αWF (t) + β(t + WF (t)).
More generally, the expected cost incurred by a customer who selects her arrival by sampling
7
from probability distribution G is
Z ∞
(αWF (t) + β(t + WF (t))) dG(t).
CF (G) =
−∞
A strategy profile is a collection {Gs (·), s ∈ [0, Λ]} of probability distributions on the real
line, one for each customer s. One may now consider the resulting decision problem as a
game between a continuum of players ([15]), and define a Nash equilibrium for this game
in terms of individual cost functions. Here, we will adopt the following Nash-Wardrop
definition of an equilibrium profile, that is more useful for the purposes of our analysis.
Definition 1 An arrival profile F is an equilibrium profile if every point ta in the
support of F is a minimizer of CF (t), namely,
CF (ta ) ≤ CF (t)
for all − ∞ < t < ∞ .
In other words, on the support of F the cost CF (t) is constant and minimal.
We note that any strategy profile {Gs (·), s ∈ [0, Λ]} that sums up to an equilibrium profile
F may be considered an an equilibrium point of the game. In particular, F induces a
symmetric equilibrium strategy profile {Fs } by letting Fs (t) = Λ−1 F (t) for every s.
The next lemma helps in simplifying the cost expression under equilibrium arrival profile.
Some notation is needed to state it. Recall that Tf = Λ/µ and let
t∗ = inf{t ≥ 0 : F (t) < µt}.
This denotes the first time beyond zero that the server starts to serve at less than the full
rate µ.
Lemma 1 Under equilibrium arrival profile F ,
(i) t∗ = Tf (i.e., the server works at full rate till the last customer is served).
(ii) For t ∈ [0, Tf ],
W (t) = F (t)/µ − t.
(2)
(iii) There are no point masses in F , i.e., F (t) is absolutely continuous in t.
Proof: (i) Clearly, t∗ ≤ Tf . Suppose that t∗ < Tf . Then, F cannot be an equilibrium
arrival profile. To see this, note that at time t∗ , there is no queue and hence W (t∗ ) = 0.
8
Furthermore, since t∗ < Tf , all the customers have not arrived and hence F (t∗ ) < Λ. Thus,
the customers that arrive after t∗ can improve their cost by arriving at t∗ , providing the
desired contradiction.
(ii) Note that (2) is obvious for t < 0 as −t is the customer wait before the server becomes
active, and F (t)/µ denotes the remaining queueing delay, once the server is active. For
t ≤ Tf , (2) follows as the queue serves at full rate in the interval [0, Tf ], so that Q(t) =
F (t) − µt and W (t) = Q(t)/µ.
(iii) Suppose that F has a point mass of size λ > 0 at some t = t1 . Then any of the
customers that arrive at t1 sees on average, half (λ/2) of the customers that arrive at
t1 before her. However, by arriving at t1 − with > 0, such a customer would arrive
ahead of this bunch, thereby reducing its waiting time by λ/2µ − at least, and leaving
earlier. Clearly, for small enough this means that arriving at t1 is not optimal for such
a customer. It follows that F has no point masses, namely no discrete component. Since
F has no continuous singular component by assumption, it follows that F is absolutely
continuous. It follows from Lemma 1 that under equilibrium arrival profile F , the cost CF (t) at any
time t ≤ Tf equals
CF (t) = (α + β)F (t)/µ − αt.
(3)
Let T0 = − Λµ αβ . The equation (3) becomes independent of t for t ∈ [T0 , Tf ] if we select
F = F ∗ where F ∗ (t) = 0 for t ≤ T0 , F ∗ (t) = Λ for t ≥ Tf , and
F ∗ (t) = Λ
t − T0
Tf − T0
for t ∈ [T0 , Tf ]. In that case, (3) equals βΛ/µ for t ∈ [T0 , Tf ]. Hence, if each customer
arrives at any time on the support of F ∗ , i.e., in [T0 , Tf ], the cost is constant and equals
βΛ
.
µ
Theorem 1 F ∗ corresponds to a unique equilibrium arrival profile.
Proof: Suppose that F is an equilibrium arrival profile with support S. Then, along this
support the cost is some constant c and it is ≥ c elsewhere. Let t0 be the left boundary
of S and t1 be its right boundary. Clearly, from Lemma 1, t1 ≤ Tf . It is easy to see that
t1 = Tf , else if t1 < Tf , the customer arriving at time t1 and getting served at Tf can
improve her cost simply by arriving at T1 . Hence, c = β Λµ .
It also follows that t0 = T0 . Arrival at this point gets served at time zero and has waiting
cost β Λµ . If this were not true and if t0 < T0 , the customer arriving at that time can improve
9
her cost by arriving instead at Tf . If t0 > T0 , then the customer arriving at Tf can improve
her cost by arriving at t0 .
Note that the cost to arrival at time t ∈ S equals (α + β)F (t)/µ − αt. For this cost to be
constant along S, S must be an interval [T0 , Tf ] (note that there cannot be an open interval
where F is constant and the (α + β)F (t)/µ − αt is also constant). Clearly F cannot have
jumps in the interval [T0 , Tf ]. Therefore, it is absolutely continuous and hence,
F (t) =
Λ
αµ
t+β
α+β
α+β
for t ∈ [T0 , Tf ]. For t > Tf , F (t) = Λ and for t < T0 , F (t) = 0. Therefore, F = F ∗ is the
unique arrival profile. Remark 3 It is worth pointing out here that while the equilibrium arrival profile is
unique, the equilibrium strategy {G∗s , s ∈ [0, Λ]} need not be. Nevertheless, if we consider symmetric equilibrium strategies, i.e., G∗s is same for all players, such a symmetric
equilibrium strategy is also unique and given G∗s (t) = F ∗ (t)/Λ.
3.1
Price of Anarchy
At the socially optimal solution, each customer selects a distribution in such a way so
that the total cost to all the customers is minimized. We first develop a lower bound on
this cost and then propose a distribution that achieves this. To get a lower bound, note
that the smallest value of W (t) is zero for all t. Also note that the total time to service
is minimized if the server serves at the fastest possible rate. That is, it functions at full
rate µ starting at time zero and serves all the customers by time Tf . Then, the average
time to service for each customer is Tf /2. So, the lower bound on the overall cost is
βTf /2 = Λβ/(2µ). It is easy to see that arrival profile F (t) = t/Tf for 0 ≤ t ≤ Tf achieves
this lower bound. Under this arrival profile, the queue always equals zero and there is no
waiting so W (t) = 0. Furthermore, under F , the service is provided to all at the fastest
possible rate: All customers are served by time Tf , so the average time to service of all
customers is also minimized.
Recall that the cost associated with the unique equilibrium profile F ∗ equalled Λβ
. Thereµ
fore, the price of anarchy defined as the ratio of maximum cost over all equilibria with the
global welfare solution, equals 2.
10
3.2
When Order of Service Matters
In our cost structure, we have thus far considered two components: The cost of time at
which service is received plus the cost of waiting in a queue. In many settings, such as at
the concert theater, while the cost of waiting is appropriate, the other component of cost
is better modeled as proportional to the number of customers that have received service
before the tagged customer gets served. Fortunately, that leads to only minor changes in
our fluid model.
To see this note that in the fluid model this change corresponds to replacing the cost
β(t + W (t)) + αW (t)
with
βF (t) + αW (t).
(4)
Again, it may be argued as in Lemma 1 that in equilibrium F it holds that t∗ = Tf , and
therefore W (t) = F (t)/µ − t. Thus, the cost (4) equals
F (t)
(α + β̂) − αt,
µ
where β̂ = βµ. This differs from the previously analyzed cost function only in that β̂
replaces β. Thus, all our previous results hold for this model as well after making this
substitution. In particular, the PoA remains 2.
4
The Multiclass Problem
We now turn to consider the more general model where customers are heterogeneous in
terms of their cost parameters. Accordingly, we divide the customer population into a
number of classes, each characterized by its own cost parameters. We consider here the
case of a finite number of customer classes. We note that the analysis may be similarly
extended to model a continuum of classes, which is omitted from the present paper due to
space limitations.
4.1
The Model
Let I = {1, 2, . . . , I} denote the set of customer classes. As before, we consider a fluid
model that represents a continuum of infinitesimal customers, arriving at a service facility
11
with service rate µ that becomes active at time t = 0. For each class i ∈ I, let Λi denote
the total workload carried by its members. Thus, all users of class i could be served within
P
Λi /µ time units. Denote Λ = i Λi .
Let Fi denote the arrival profile for class i. It is a positive measure on the real line with
total mass Λi . An arrival profile is the collection {Fi } of arrival profiles for all classes. The
P
sum F (t) = i Fi (t) is the aggregate arrival profile.
P
As before, given the aggregate arrival profile F = i Fi , the queue size Q(t) and waiting
time W (t) of a (possibly virtual) customer that arrives at time t is well defined and so is
her cost
Ci (t) = αi W (t) + βi (t + W (t)),
where αi > 0 and βi > 0 are class specific parameters. Thus, as for the single class, the
waiting time of a customer who arrives at time t and encounters a queue Q(t) before her
will be W (t) = Q(t)/µ = min{0, −t}, so that she completes her service and leaves the
system at tc = t + W (t) = Q(t)/µ + max{0, t}.
Equilibrium arrival profile may be defined analogously.
Definition 2 Let {Fi } be an arrival profile with associated waiting time function W (t) and
cost functions Ci (t). Then {Fi } is an equilibrium profile if for every class i, every point
ta in the support of Fi is a minimizer of Ci (t), namely,
Ci (ta ) ≤ Ci (t)
4.2
for all − ∞ < t < ∞ .
The Equilibrium Profile
We proceed to identify explicitly the equilibrium arrival profile. To that end, define the
cost ratio parameters
αi
.
mi =
αi + βi
Let us reorder the class indices in increasing order of mi , so that mi ≤ mi+1 . We will
assume for simplicity that all the cost ratio parameters mi are distinct. When this is not
the case, one can simply unify customer classes that have identical mi ’s, and all the results
of this section essentially hold.
Theorem 2 Suppose m1 < m2 < · · · < mI . Then the equilibrium profile {Fi } exists, is
unique, and specified as follows: Let T0 < T1 < · · · < TI be an increasing sequence of time
12
instants defined by
TI = Λ/µ,
Ti−1 = Ti −
Λi
, i = 0, 1, . . . , I .
µmi
(5)
Then, Fi corresponds to a uniform distribution on [Ti−1 , Ti ] with density µmi , namely
Fi0 (t) = µmi 1{Ti−1 ≤ t < Ti }
(6)
We proceed to prove this result. To begin with, we observe that Lemma 1 and its proof
remain unchanged in the multiclass case. Thus, under any equilibrium profile {Fi }, the
server operates at its full rate µ from time 0 till the last customer is served. Hence all
customers are served by time Tf = Λ/µ. Furthermore, a customer that joins the queue at
time t will leave it at time
tc = F (t)/µ.
Therefore, the cost function for a class i arrival at t,
Ci (t) = αi (tc − t) + βi tc = (αi + βi )tc − αi t
= (αi + βi )
F (t)
− αi t
µ
(7)
The next lemma establishes the relationship between the arrival times of the different
classes in equilibrium.
Lemma 2 Let {Fi } be an equilibrium profile.
(i) If an interval (t1 , t2 ) belongs to the support of Fi (t), then
Fi0 (t) = µmi for t ∈ (t1 , t2 )
(ii) Let i and j be two class indices so that mi < mj . Then all arrivals of class i occur
before those of class j.
The following lemma is useful for proving Lemma 2.
P
Lemma 3 Let {Fi } be an equilibrium profile, and denote F = i Fi . Then there are no
gaps in the aggregate arrival profile: That is, F (t2 ) − F (t1 ) > 0 for all t2 > t1 such that
0 < F (t1 ) < Λ.
13
Proof: Suppose, to the contrary, that there are no arrivals on (t1 , t2 ). By our assumptions
on t1 there are some arrivals both before and after this interval. Since the server operates at
full rate over (t1 , t2 ), it follows that the last customer to enter before t1 will not get served
before t2 . Therefore, by arriving just before t2 , this customer will reduce her waiting time
while leaving at the same time as before, thereby improving her cost. Thus, this arrival
profile cannot be an equilibrium profile.
Proof of Lemma 2: (i) By the equilibrium definition, it follows that Ci (t) is constant on
(t1 , t2 ). From Lemma 1 it easily follows that each Fi is absolutely continuous so it admits
a density that we denote by Fi0 (t).
Noting (7), it follows by differentiation that on that interval,
αi
= µmi .
Fi0 (t) = µ
αi + βi
(ii) Suppose there are classes i and j with mi < mj such that some class j arrivals arrive
in some interval (t1 , t2 ) just before class i arrivals in some interval (t2 , t3 ) with t1 < t2 < t3 .
That there will be non-zero arrivals in each of these two intervals is given by Lemma 3.
Let us compare the cost incurred by a class j arrival on these two intervals. For t ∈ (t1 , t2 ),
Cj (t) is constant (by definition of the equilibrium) and equals Cj (t2 ) (by continuity). Now,
from item (i) we know that F 0 (t) = µmi on (t2 , t3 ), hence on that interval,
F (t)
d
0
− αj t
(αj + βj )
Cj (t) =
dt
µ
F 0 (t)
= (αj + βj )
− αj = (αj + βj )mi − αj
µ
= (αj + βj )(mi − mj ) < 0 .
This implies that the cost Cj (t) is strictly smaller on (t2 , t3 ) than on (t1 , t2 ), which shows
that the latter interval cannot be in the support of Fj at equilibrium, contrary to our
assumption.
Proof of Theorem 2: To establish Theorem 2, we first show that an equilibrium profile
must have the indicated form. From Lemma 2(ii) it follows that the arrivals of the different
classes are ordered in increasing order of their mi parameters. Now, from Lemma 3 it
follows that the arrivals of each class i are supported on a single interval [τi , Ti ], and that
these intervals are contiguous so that τi = Ti−1 . From Lemma 2(i) we see that the arrival
profile of each class i on its interval [Ti−1 , Ti ] is uniform with rate µmi . Computing the
overall arrival volume on that interval gives µmi (Ti −Ti−1 ) = Λi , which implies the recursive
relation in (5). Finally, TI = Λ/µ follows from Lemma 3, as already indicated.
14
It is now a simple matter to verify that the indicated arrival profile is indeed an equilibrium
profile. Clearly, the cost Ci (t) is constant on [Ti−1 , Ti ] by construction. Moreover, arguing
as in the proof of Lemma 2, it is readily verified that Ci0 (t) > 0 for t > Ti and Ci0 (t) < 0
for t < Ti−1 , thereby establishing that the cost Ci (t) is indeed minimized on the support
[Ti−1 , Ti ] of Fi .
We end this subsection with a few observations regarding the equilibrium profile. The
P
aggregate arrival profile F (t) = i Fi (t) can be expressed more explicitly as follows. F (t)
is piecewise linear, with slope µmi on [Ti−1 , Ti ]. The times Ti are given by
Ti = Λ/µ −
I
X
Λj
.
µm
j
j=i+1
(8)
At these times,
F (Ti ) = Λ −
I
X
j=i+1
Λj =
i
X
Λj
(9)
j=1
with linear interpolation on [Ti−1 , Ti ] at slope µmi (see Figure 1). Note that T0 < 0 (since
mi < 1), so that arrivals start before t = 0 as in the single class case. Further, the
aggregate arrival profile is convex for t ≤ TI , meaning that the arrival rate is increasing
in time, reaching its peak towards the end of the service period. Still, the queue length is
strictly decreasing beyond t = 0 (which again follows since mi < 1.) Finally, arrivals are
i
ordered in increasing order of mi = αiα+β
, or equivalently in increasing order of αβii which
i
indicates the relative cost they attribute to waiting over being late.
4.3
Price of Anarchy
We now turn to compute and bound the Price of Anarchy (PoA) for the multiclass model2 .
For that purpose we first compute the social cost at equilibrium, Jeq , followed by the
optimal social cost Jopt .
Computing Jeq : The social cost is defined as the sum of all costs of all customers, at the
given arrival profile. Consider the equilibrium arrival profile of the previous subsection.
Since the equilibrium cost Ci (t) is the same for all members of each class, say Ci , we obtain
X
Jeq =
Λi Ci
(10)
i
2
Due to space constraints, we omit all proofs of the claims in this subsection
15
Figure 1: The cumulative distribution of the aggregate arrival profile in equilibrium
The cost Ci may be computed in any point in [Ti−1 , Ti ]. Picking Ti , we get
F (Ti )
− αi Ti
µ
Substituting Ti and F (Ti ) from (8) and (9), one may obtain after some algebraic manipulations the following simplified expression:
Ci = Ci (Ti ) = (αi + βi )
Jeq =
I
1 X
βi βj
Λi Λj αi min{ , }
µ i,j=1
αi αj
We note that this expression is independent of ordering of the classes.
The Optimal Social Cost: The optimal social cost Jopt is obtained by optimizing the arrival
times and server allocation for all customers. Here there would be no queues, as each
customer can arrive exactly when her turn to be served arrives. It may then be seen through
a simple interchange argument that the optimal ordering of arrivals between classes is in
decreasing order of βi . This defines completely the arrival profile and associated cost, and
after some algebraic manipulations we obtain
Jopt
I
1 X
=
Λi Λj min{βi , βj }
2µ i,j=1
The equations derived above imply the following explicit expression for the PoA:
PI
βi βj
i,j=1 Λi Λj αi min{ αi , αj }
4 Jeq
PoA =
= 2 PI
Jopt
i,j=1 Λi Λj min{βi , βj }
16
(11)
(12)
As we will see below, the PoA ranges around its single-class value of 2. We proceed to
derive some bounds on this value. Essentially, we will be interested in bounds that depend
only on the ranges of the cost parameters (αi and βi ) but not on the relative size (Λi ) of
the customer classes. We start with some special cases, where only one type of parameters
varies across classes.
Proposition 1
(i) Identical wait sensitivities. Suppose αi ≡ α0 : the wait sensitivities are identical for
all customer classes. Then PoA = 2.
(ii) Identical lateness sensitivities. Suppose βi ≡ β0 : the lateness sensitivities are identical
for all classes. Then PoA ≤ 2, and
αmin
αmin
−1
≥1+
,
(13)
PoA ≥ 2 − (1 − I ) 1 −
αmax
αmax
where αmax = maxi αi , αmin = mini αi , and I is the number of classes.
Item (i) of the last proposition is evidently an exact extension of the PoA result for the
single-class case, giving the same value of 2. Regarding (ii), we first note the upper bound
of 2 is strict unless all the αi ’s are equal as well. Thus, in this case, diversity in the waiting
sensitivities of the customers actually improves the PoA compared to the single class case.
As for the lower bound, for two user classes (I = 2) with α1 < α2 it reads
α1
PoA ≥ 1.5 + 0.5
α2
We observe that this bound is tight, and is achieved when Λ1 = Λ2 .
We now turn to consider the general case, when both sets of cost parameters may vary across
customer classes. The following set of bounds is obtained simply by bounding separately
the ratios of each pair of corresponding terms in the numerator and denominator of (12).
Proposition 2 Let Gmax = maxi,j G(i, j) and Gmin = mini,j G(i, j), where
β
G(i, j) =
(αi + αj ) min{ αβii , αjj }
2 min{βi , βj }
Then 2Gmin ≤ PoA ≤ 2Gmax . Consequently,
αmax βmax
PoA ≤ 1 + min
,
αmin βmin
αmin βmin
PoA ≥ (1 +
)
αmax βmax
17
(14)
(15)
Equation (14) provides, in particular, an upper bound on the PoA in terms of the β
parameters only. In fact, a tighter bound of this form may be derived through somewhat
refined analysis. This bound also points to the “worst case” conditions in terms of the PoA
when the (βi ) parameters are given.
q
Proposition 3 PoA ≤ 1 + ββmax
.
min
We note that the bound of the last proposition is tight, in the sense that for any set of βi ’s,
the bound is satisfied with equality for some (αi , Λi ) parameters. Indeed, as implied by the
proof (which is omitted here), setting the βi ’s in increasing order, equality is obtained for
p
Λ2 = · · · = ΛI−1 = 0, Λ1 /ΛI = βI /β1 , and αI /α1 = βI /β1 (cf. (12)).
5
Managing the Price of Anarchy
In this section we discuss some system management tools that may be used to reduce the
price of anarchy. For simplicity, we consider setting of a single customer class, although
the generalization to multi-class may be carried out in a similar manner.
We first consider the case where PoA may be controlled by segmenting the population so
that certain proportions are served only after specified thresholds. This has applications
in numerous settings. For instance, when large number of candidates are to be interviewed
by an organization, often they are segmented and are asked to report at different time segments. Visa and immigration centers often through online scheduling assign separate time
bands to customers coming for an appointment. We also discuss performance degradation
that may occur if sub-optimal choices are made. This is of obvious practical importance as
it is difficult to make optimal decisions given inherent model uncertainty.
We then briefly discuss how improved PoA may be obtained by assigning varying priorities
to different segments of population. One popular application that closely approximates
this is in airplane boarding where economy customers are assigned different priority based
on their seat location. Finally, we discuss how better price of anarchy may be obtained
through differential pricing by charging a tariff to customers that are served early. When
done optimally, this can be quite effective in controlling PoA. We also discuss performance
degradation that may occur with suboptimal pricing.
In the above three settings, it is easy to quantify the reduction in price of anarchy as function
of the population segments created. In particular, we see that when done optimally, the
18
price of anarchy equals 1 + 1/n where n denotes the number of population segments.
Without loss of generality we take Λ = 1 in this section.
5.1
Service Delayed for Some Customers
To convey the main points simply, we focus primarily on the case where population is
divided into two segments. Specifically, consider the case where (1 − a) proportion of the
population is allowed to be served only after time τ̂ > 0. Call this the second population.
The first population corresponds to the proportion a that is allowed to be served at any
time t ≥ 0. We allow the second population to queue up before time τ̂ , so that after time
τ̂ they join the end of the queue of population 1 customers at the service facility (if any)
and are served after them. Within the same population the service is always in the order
of customer arrival. After time τ̂ , customers from both the populations join at the end of
the existing queue at the service facility and are served in the order they arrive.
We first consider the case τ̂ = µa . This is a critical point as population 1 completes its
service requirements at time µa . We do not discuss the case τ̂ > µa separately as in this
case the queue is empty in the interval τ̂ − µa and the analysis is trivial. The case τ̂ < µa
is analyzed in some detail as it provides interesting insights on how variedly customers
may behave as τ̂ decreases. The key observation is that there is a certain phase change in
customer behavior at τ̂ = τ̂ ∗ , where
τ̂ ∗ =
a
β
− (1 − a) .
µ
αµ
(16)
While for τ̂ ∗ < τ̂ < µa , there exists a unique equilibrium solution and the aggregate arrival
profile varies with τ̂ , for τ̂ < τ̂ ∗ there may be multiple equilibria, the aggregate arrival
profile is independent of τ̂ and is identical to the unconstrained case.
5.1.1
τ̂ =
a
µ
Under this scheme, the unique equilibrium is easily seen to correspond to both the populations blissfully unaware of the other, the first population arrives as if the second does
not exist and the server facility opens at time 0, the second population arrives as if the
server facility opens at time a/µ and queues up appropriately before time a/µ. Specifically,
the first population has a proportion of customers that arrive uniformly between the inβa a
terval [− αµ
, µ ] and the second population has (1 − a) proportion of customers that arrive
19
Figure 2: Equilibrium queue length profile for the two populations. Population 1 comprises
a proportion and is served in the interval [0, a/µ]. Population 2 comprises (1−a) proportion
and is allowed service after time a/µ, although it starts queueing from time τ̂ ∗ onwards.
αµ
uniformly between [τ̂ ∗ , µ1 ]. Both arrive at rate (α+β)
in their respective arrival intervals so
that the cost incurred by arrivals in each population is constant independent of the arrival
times. See Figure 2 for an illustration.
Then, a customer from the first population has no incentive to arrive outside the interval
βa a
[− αµ
, µ ] where the cost would be higher. Similarly, the customer in second population
arriving at any time in the interval [τ̂ ∗ , µ1 ] has a constant cost and a higher cost outside
this interval.
The cost incurred by a customer in the first population equals βa/µ, and that by a customer
in second population equals β/µ.
The overall cost, since population 1 has proportion a and population 2 has proportion 1−a,
equals
β 2
(a + (1 − a)).
µ
The social optimal corresponds to zero waiting and the associated overall cost equals
The PoA then equals
2(a2 + (1 − a)).
20
β
.
2µ
This is minimized at a = 1/2 where the PoA equals 3/2.
Hence, to achieve optimal PoA, when the population is segmented into two parts, it is best
to schedule half the population to come half the total serving time later.
5.1.2
τ̂ <
a
µ
The following proposition summarizes this setting for different values of τ̂ . Let m denote
α
the ratio (α+β)
.
Proposition 4
1. For τ̂ ∗ < τ̂ < µa , there exists a unique equilibrium where the first population arrives
uniformly between the interval
[−
βa
a
− ( − τ̂ ), τ̂ ]
αµ
µ
at rate µm and the second population arrives uniformly between [τ̂ ∗ , µ1 ] at rate µm.
The PoA increases linearly from 2(a2 + (1 − a)) to 2 as τ̂ decreases from µa to τ̂ ∗ .
2. For τ̂ ≤ τ̂ ∗ , under an equilibrium, population 1 arrives uniformly between
[−
β ∗
, τ̂ ]
αµ
at rate µm and population 2 arrives uniformly between [τ̂ ∗ , µ1 ] at rate µm. Here,
multiple equilibria may exist, and in each equilibrium, each customer in either class
incurs a cost βµ . Hence, PoA equals 2.
Let c1 = α( µa − τ̂ ) + β µa and c2 = β/µ.
Proof: First consider τ̂ ∗ < τ̂ < µa . We first argue that the specified arrival profile of
the two populations is in equilibrium. To see this note that each customer in population
1 incurs a constant cost equal to that incurred by the last customer of this population
arriving at time τ̂ and served at time µa , i.e., c1 .
The cost incurred by each customer in population 2 equals c2 > c1 . Therefore, customers
in population 1 have no incentive to come after time τ̂ . They clearly have no incentive to
βa
− ( µa − τ̂ ). Similarly, population 2 has no incentive to come outside the
come before − αµ
the specified intervals.
21
To see that this equilibrium is unique, note that as before, under any equilibrium, the server
will serve at a full rate till time 1/µ. Clearly, the last customer to be served in equilibrium
will arrive at time 1/µ and incur the cost c2 . She cannot be from population 1, as then she
has the option of arriving at time τ̂ and be served by at most time a/µ. That is, her cost
in equilibrium must be bounded from above by c1 < c2 .
Hence, population 2 customer is the last one to arrive and in equilibrium the cost incurred
by each customer of population 2 equals c2 . Clearly, population 1 cannot arrive after
time τ̂ and incur cost less than c2 as then a customer from population 2 can replicate
this to lower her cost. Hence, since each customer in population 1 has a constant cost,
, τ ] at rate µm for some τ ≤ τ̂ .
this population arrives uniformly in an interval [τ − a(α+β)
µα
Again, if τ < τ̂ , the last customer of this population can improve her cost by coming at
τ̂ , so τ = τ̂ . In particular, cost incurred by population 1 customer equals c1 , and they are
served uninterruptedly till time a/µ. From population 2’s viewpoint, then, in equilibrium
the queue opens at time a/µ, and hence in equilibrium it must follow the profile specified
in the proposition.
It is easily seen that PoA increases linearly from 2(a2 + (1 − a)) to 2 as τ̂ decreases from
a
to τ̂ ∗ .
µ
Now consider the case τ̂ ≤ τ̂ ∗ . First note that under the strategy specified in this proposition, each customer from both the populations incurs cost c2 , and cannot improve this by
arriving at another time.
To see that all equilibria must have cost c2 , first note that in equilibrium the cost incurred
by either population cannot be more than c2 , the cost incurred by the last customer arriving
at 1/µ.
Now suppose that τ̂ < τ ∗ so that c1 > c2 . If all of population 1 arrives by time τ̂ , the
cost incurred by its last customer (who will be served at time a/µ and will need to wait at
least a/µ − τ̂ ) is greater than c1 . Hence, this cannot be in equilibrium and some customers
must arrive after time τ̂ . These must have the same cost as population 2 customers in
equilibrium. Thus, equilibrium cost for each customer must equal c2 .
Finally, consider τ̂ = τ ∗ so that c1 = c2 . Then, even if all customers of population 1 arrive
by τ̂ , their cost must be constant and equal to that of their last customer that has to arrive
at τ ∗ in equilibrium. So, there cost must equal c2 . Therefore, in all equilibria, the cost
incurred by each customer equals c2 . Hence, PoA equals 2.
22
Remark 4 Note that in the latter case multiple equilibria with same cost exist in the
sense that any strategy where
β
τ̂ + αµ
β
τ̂ ∗ + αµ
β
arrive uniformly at rate µm in the interval [− αµ
, τ̂ ],
and the remaining population 1 and population 2 customers arrive uniformly at rate µm
in the interval [τ̂ , µ1 ] is also an equilibrium strategy as the cost incurred by each customer
in either population is βµ .
5.1.3
Generalization to multiple thresholds
1
for a = 1/2, we obtained the optimal PoA of
Note that when we set τ̂ optimally at 2µ
3/2. This generalizes so that if we restrict m
proportion of people to come after time n−m
n
nµ
time for m = 1, 2, . . . , n − 1 then the equilibrium cost for customers getting served in a slot
m m+1
, nµ ) equals
( nµ
(m + 1)β
.
nµ
Again, this is 1/n proportion of the population so that the total cost equals
2
β 1
( + + ... + 1)
nµ n n
or
β(n+1)
2nµ
5.2
so that the PoA equals (n + 1)/n and converges to 1 as n → 1.
Priority Queueing
as in Subsection 5.1.3 is through dividing the
Another way to achieve PoA equal to n+1
n
population into n separate segments and assigning different priorities to them. Specifically,
suppose that the population is divided into n segments with (ai : i ≤ n) denoting the
respective proportions (the cost function is identical for each segment). The population
segment with lower index is given priority over the segment with higher index. Then, in
equilibrium customers arrive in disjoint intervals, customers of segment 1 arrive first uni1 a1
formly in the interval [− βa
, ] and are served by the server in the interval [0, aµ1 ]. Similarly,
αµ µ
P
aj
βai Pj
ai
customers of segment j ≥ 2 arrive uniformly in the interval [ j−1
i=1 µ − αµ ,
i=1 µ ] and
P
Pj ai
ai
are by the server in the interval [ j−1
i=1 µ ,
i=1 µ ].
Pj ai
The cost incurred by segment i equals β i=1 µ so that overall price of anarchy equals
" n
#
j
X X
2
aj (
ai ) .
j=1
i=1
23
Through simple optimization, it can be seen that this is minimized by setting aj =
as in Subsection 5.1.3.
each j so that PoA equals n+1
n
5.3
1
n
for
Reduction in PoA through Charging Tariffs
Recall that in Section 5.1 in the two population setting, we obtained the best PoA when
we divided the populations in equals parts and allowed the second population to come
1
1
after time 2µ
. Then, the cost to each customer in the first population was 2µ
less than
that of customers in the second population. This suggests a procedure for implementing
differential pricing.
For brevity, we restrict our discussion to the case where customers joining the service facility
1
have to pay a constant tariff p while the customers joining the service
queue by time 2µ
facility queue after this time pay no tariff. We refer to the former as population 1 and
latter as population 2. We assume here that demand of one unit is fixed and is unaffected
by the pricing strategy of the service provider. Again, we allow population 2 to queue up
1
1
before time 2µ
separately and join at the end of service facility queue at time 2µ
. In this
case they are served after population 1 customers at the service facility queue at that time,
if any, and in their order of arrival amongst population 2. We further assume that the
tariff collected is returned to the society so this does not enter into the price of anarchy
calculations. We now discuss different scenarios depending upon the value of p. The proofs
of Propositions 5 and 6 are not central to our analysis and are given in the appendix.
5.3.1
p=
β
2µ
β
1
, 2µ
] at rate µm, and
In this scenario, the first population arrives uniformly between [− 2αµ
β
1
1
the other between [ 2µ − 2αµ , µ ] at the same rate. The cost incurred by both the populations
β
β
is βµ : For the first population it is 2µ
from waiting and time to service and another 2µ
from
the tariff for coming early.
Thus, a customer is indifferent to coming as part of population 1 or 2. The revenue collected
β
by the service provider from tariffs equals 4µ
. The PoA, as before, equals 3/2. See Figure
3 for an illustration of this scenario.
5.3.2
β
p = (1 + c) 2µ
, c>0
The following proposition summarizes this setting for different values of c.
24
Figure 3: The dotted line denotes the queue profile before differential pricing. After differential pricing the darkened line denotes the queue profile of population 1 that pays β/(2µ)
more than population 2 whose queue profile is shown using the lighter line. The cost to
customer joining either of the two populations equals β/µ.
Proposition 5
1. Under unique equilibrium, ( 12 − 4c ) proportion of customers arrive as population 1, for
c ≤ 2, at rate µm, uniformly between
β 1 c 1 1 c
− ( − ), ( − ) ,
αµ 2 4 µ 2 4
and ( 12 + 4c ) proportion arrive as population 2 at rate µm uniformly between
β
c 1
c
1
−
(1 + ), +
.
2µ 2αµ
2 µ 4µ
For c ≥ 2, all customers come as population 2 as for c = 2.
2. Furthermore, for c ≤ 2, PoA equals
=
3 c(1 + c)
+
.
2
4
For c > 2 it equals 3.
See Figure 4 for an illustration.
5.3.3
β
p = (1 − c) 2µ
, 0<c<1
The following proposition summarizes this setting for different values of c.
25
(17)
Figure 4: The dotted line denotes the queue profile before differential pricing. The darkened
line denotes the queue profile of population 1 that pays β(1 + c)/(2µ) more than population
2 whose queue profile is shown using the lighter line. The population 1 is served till
c
c
1
− 4µ
and population 2 is served till τ̃3 = µ1 + 4µ
. The cost to customer joining either
τ̃1 = 2µ
β
c
of the two populations equals µ (1 + 4 ).
Proposition 6
1. For 0 ≤ c ≤ 1, under unique equilibrium, proportion
as population 1 at rate µm, uniformly between
1
β
(1 + c),
−
.
2αµ
2µ
In addition, proportion
uniformly between
1
2
−
βc
2(α+β)
1
2
+
βc
2(α+β)
of customers arrive
of customers arrive as population 2 at rate µm,
1
β
1
−
(1 − c),
.
2µ 2αµ
µ
2. Furthermore,
P oA =
3 c(α + βc)
+
.
2
2(α + β)
(18)
This equals 3/2 at c = 0 and 2 at c = 1.
See Figure 5 for an illustration.
β
Note that for tariff 0 ≤ p ≤ 2µ
, the cost to each customer remains fixed at βµ while this had
β
increased for p > 2µ
. It is easily seen that by having n − 1 separate tariffs so that customers
i i+1
served in the interval ( nµ
, nµ ) for (i = 0, 1, 2, . . . , n − 1) are charged amount βµ n−i−1
, we
n
n+1
can achieve PoA equal to n as in Subsection 5.1.3.
26
Figure 5: The dotted line denotes the queue profile before differential pricing. The darkened
line denotes the queue profile of population 1 that pays β(1−c)/(2µ) more than population
2 whose queue profile is shown using the lighter line. The population 1 is served till
βc
1
τ̆1 = 2µ
+ 2µ(α+β)
. The cost to customer joining either of the two populations equals βµ .
6
Conclusion
In this paper we considered the queueing problem that may arise in settings such as concert
and movie theaters, cafeterias, DMV offices, Black Friday shopping queues etc., where a
large number of customers may queue up before a facility that opens for service at a
particular time. The customers strategically select their arrival time distributions to tradeoff waiting time in queue with costs due to late arrival. We developed a queueing framework
for this problem for which we identified the fluid limit. We observed that the fluid limit
allows a great deal of tractability in analyzing the strategic arrival problem faced by each
customer. We identified the unique arrival profile for each customer class in equilibrium,
and showed that the price of anarchy equals 2 in the single-class model while it varies
around this value in the multiclass case. We further discussed structural changes in the
queueing discipline and simple pricing schemes that may reduce the price of anarchy.
As part of future work, we plan to study the equilibrium properties of the fluid model under
more general cost functions as well study the model introduced here under the diffusion
limit. Extension to multi-server queueing networks would also be of interest in many
applications particularly communication networks. We hope that this analysis motivates
further research in strategic analysis of queueing systems.
27
7
Appendix: Some Proofs
Proof of Proposition 5: Note that in an equilibrium, both populations will arrive in
disjoint intervals at rate µm. Suppose that the population 1 arrives uniformly between
1
and τ̃0 < 0 (recall that service begins at time zero). The second
[τ̃0 , τ̃1 ], where τ̃1 ≤ 2µ
1
population arrives uniformly between times [τ̃2 , τ̃3 ] for τ̃3 > µ1 and τ̃2 < 2µ
(recall that
1
service for this population begins at 2µ ). Since the cost incurred by the two populations is
β
+ β τ̃1 = β τ̃3 . Since the total service allocated is for time µ1 , we
the same, we have (1 + c) 2µ
1
have τ̃1 + (τ̃3 − 2µ
) = µ1 .
c
1
c
It follows that τ̃3 = µ1 + 4µ
, τ̃1 = 2µ
− 4µ
, and the cost incurred by each customer equals
β
c
(1 + 4 ). The proportion of customers coming in as population 1 equals µτ̃1 = ( 21 − 4c ).
µ
τ̃0 and τ̃2 can be easily seen to be as specified in the proposition since the arrival rates for
each population are known.
To compute PoA, note that the revenue from population 1 equals β τ̃1 ( 12 − 4c ) = βµ ( 21 − 4c )2 .
The revenue from population 2 equals β τ̃3 ( 21 + 4c ) = βµ (1 + 4c )( 12 + 4c ) , so that (17) follows.
Proof of Proposition 6: It can be argued as in the proof of Proposition 4 that population
1
1
1 arrives uniformly between [τ̆0 , 2µ
], at rate µm and is served till time τ̆1 , for some τ̆1 > 2µ
and τ̆0 < 0. Also, population 2 arrives uniformly between [τ̆2 , µ1 ] at rate µm for some τ̆2 .
Note that the cost incurred by the two populations is the same and that specifies τ̆1 through
the equation
1
β
β
+ α(τ̆1 −
) + β τ̆1 = ,
(1 − c)
2µ
2µ
µ
or
1
βc
τ̆1 =
+
.
2µ 2µ(α + β)
The proportion of customers coming in as population 1 equals
µτ̆1 =
1
βc
+
.
2 2(α + β)
β
It is then easily inferred that τ̆0 = − 2αµ
(1 + c). Similarly, since 12 −
β
1
population is of type 2, we can evaluate that τ̆2 = 2µ
− 2αµ
(1 − c).
βc
2(α+β)
proportion of
We now compute the PoA. Note that each customer in population 1 incurs a cost
α(τ̆1 −
1
β
) + β τ̆1 =
(1 + c).
2µ
2µ
28
Each customer in population 2 incurs cost βµ . From this we can determine the total cost
and after algebraic manipulations conclude that (18) holds.
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