1
Equilibrium and socially optimal arrivals to a single
server loss system
Liron Ravner and Moshe Haviv
Department of Statistics and the Center for the Study of Rationality
Hebrew University of Jerusalem
Abstract—We study a single server model with no queue and
exponential services times, in which service is only provided
during a certain time interval. A number of customers wish
to obtain this service and can choose their arrival time. A
customer that finds a busy server leaves without being served.
We model this scenario as a non-cooperative game in which the
customers wish to maximize their probability of obtaining service.
We characterize the Nash equilibrium and the price of anarchy,
which is defined as the ratio between the optimal and equilibrium
social utility. In particular, the equilibrium arrival distribution
has an atom at zero, a period with no arrival and is continuous
on some interval until the closing time. We further generalize
our analysis to take into account uncertainty regarding the
population size, i.e. a game with a random number of customers.
In the special case where the population size follows a Poisson
distribution, we show that the continuous part of the distribution
is uniform, which is not the case in general. Finally, we show
that the price of anarchy is not monotone with respect to the
population size; but rather uni-modal with values close to one
for small and large populations.
I. I NTRODUCTION
Many congestion scenarios take place over a fixed interval
of time, and thus require transient rather than steady state
analysis. The reason for this could be because the service
is only helpful at a certain time, such as transportation or
communication networks which are mainly utilized in certain
hours of the day. Another possibility is that the server itself
only operates during certain hours, for example a bank or a
call center. Customers who require these kind of services are
likely to select their time of arrival strategically with the goal
of minimizing costs, such as waiting or tardiness. If the server
has a limited queue buffer, which is often the case in healthcare and telecommunications applications, then some of the
customers may not receive service at all because the buffer is
full upon their arrival. In this case we suggest a cost function
that is binary: success at obtaining service, or failure. The
goal of this research is to model this phenomenon as noncooperative game in which the customers try to maximize their
probability of obtaining service by strategically selecting their
arrival time. We analyse the properties of this game, namely
the Nash equilibrium and the social optimization problem. In
this paper we analyse a single server model with no queue
buffer, but future research can extend our analysis to a multiserver queue with a limited capacity buffer.
Queueing models with strategic arrival of customers are
often limited to decisions such as whether or not to join, which
queue to join or whether to renege or not after a certain period
of waiting. A detailed survey of such models is presented by
Hassin and Haviv in [2]. The first to analyse a model where
the customers can decide the exact time of their arrival were
Glazer and Hassin in [1]. They assume that the number of
customers seeking service follows a Poisson distribution and
that the customers incur linear waiting costs, and analyse the
mixed strategy equilibrium properties. In this model customers
are allowed to arrive and wait before the opening time, this
assumption was later removed by Hassin and Kleiner in
[3]. In addition to waiting costs, customers may also have
tardiness costs. This cost structure was examined, among
others, by Haviv in [5]. A Poisson population is assumed and
equilibrium analysis is presented for several restrictions on
the time interval that customers are allowed to arrive in, along
with a comparison to a fluid approximation model. Additional
analysis of the tardiness model is carried out by Juneja and
Shimkin in [10], where they generalize the model to any
distribution of the population size. They further prove that
under very general conditions there is a unique symmetric
Nash equilibrium, that converges to the fluid solution when
the population size increases (with the appropriate scaling of
the service rate). The fluid approximation enables analysis of
more complex models such as heterogeneous customers by
Jain, Juneja and Shimkin in [9] and a network of queues by
Honnappa and Jain in [8]. Mazalov and Chuiko [12] study
a single server model with no queue buffer and customers
that have a time sensitivity function which determines their
utility. Ravner [16] introduced a model in which the utility of
the customers is dependent on their order of admittance. An
experimental study of the choice of arrival times was carried
out by Rapaport et. al in [15], and compared to the equilibrium
of a discrete time model with fixed service times.
Another natural question that arises in this setting is how
can a central planner optimize the arrival policy so that
as many customers as possible are admitted into service?
This question is of a scheduling nature, and much of the
literature on it is orthogonal to the game theoretic approach.
The optimization of arrival times to a queueing system,
with the goal of minimizing waiting costs for example, is
typically cumbersome and requires heuristic or approximation
methods. Pegden and Rosenshine [14] provide an algorithm
for minimizing customer waiting time, and show that this
objective is convex for a small number of customers. In [4]
Hassin and Mendel analyse a queueing scheduling model in
2
which the customers do not always show up at their assigned
appointment time. An optimization problem that is close to
the one studied in this paper is presented in [11] by Lin and
Ross. They seek an optimal dynamic ”gatekeeper” policy for a
partially observable loss server, where the optimizer observes
the arrival times but not the state of the server. In recent years
the social optimization problem and the equilibrium analysis
are often both carried out, which allows for analysis of the
price of anarchy, which is defined as the ratio of the social
utility in both cases. In the context of the queuing arrival time
games, this is done for example in the fluid setting in [5], and
bounds on the general case are provided in [10]. A different
approach to the scheduling problem is adopted in [3], where
they analyse the socially optimal symmetric arrival strategy,
i.e. the central planner can only assign a single strategy for all
customers to use.
In this paper we study a single server, with an exponential
service rate. A customer is only admitted into service if the
server is idle upon her arrival, and she therefore wishes to
arrive at a time that maximizes the probability of finding the
server idle. In section II we characterize the Nash equilibrium
customer behaviour for this model. We first provide analysis for a Poisson number of customers in section II-B. In
particular, we show that the equilibrium arrival distribution
has an atom at time zero and is uniform on a subset of the
server opening time interval. We proceed to characterizing the
equilibrium for a known deterministic number of customers.
In this case we formulate the equilibrium conditions as a set
of functional differential equations and provide a numerical
procedure to solve them, accompanied by several numerical
examples. In section II-F we explain how a straightforward
generalization can be made to any distribution on the number
of customers. The issue of social optimization and the price
of anarchy is addressed in section III. In particular we provide
an explicit socially optimal solution when the central planner
can use a non-symmetric scheduling policy. This also yields
an upper bound for the symmetric case, which we do not solve
here. We conclude the paper in section IV with closing remarks
and discussion of possible future research.
II. E QUILIBRIUM ANALYSIS
Suppose a single server admits customers during a finite
time interval [0, T ]. We assume service will be completed if
a customer is being served at time T . The service time of all
customers are independent and exponentially distributed with
rate µ. There is no queue and an arriving customer that finds a
busy server leaves without being served. If several customers
arrive at exactly the same time and the server is idle, then
the customer who enters service is determined by a uniform
random draw. Each customer believes that the number of other
customers is N , which may be deterministic or random. In
the latter case we assume that all customers share a common
belief that N is a non-negative random variable with a known
probability distribution. Customers select their time of arrival
ti with the goal of maximizing the probability of finding
the server idle upon arrival. Hence, this is a non-cooperative
simultaneous action game. We denote the mixed strategy of
customer i as a probability distribution with cdf Fi . The arrival
process to the server is determined by the strategy profile
F = {Fi : i ∈ {1, ..., N }}, which is the set of independent
arrival distributions chosen by all customers. The server only
has two possible states, idle and busy, that we denote by 0 and
1 respectively. Finally, we denote the probability that customer
j arriving at time t finds the server in state i ∈ {0, 1}, if all
other customers are playing strategy F −j := F \ {Fj }, by
−j
pF
(t).
i
Definition A strategy profile F is a Nash equilibrium if there
−j
exists a constant Cj such that pF
(t) ≤ Cj for any customer
0
j and any time t. Moreover, if t is in the support of Fj then
−j
pF
(t) = Cj .
0
As is necessary and sufficient for an equilibrium profile, no
customer can improve her utility by arriving at a time that
is not on her support, and they are all indifferent between
the points on their support. In some cases there may exist
pure strategy equilibria, for example in a two customer game:
one customer arriving at time zero and the second at time
T can be an equilibrium if the service rate is fast enough
with respect to the closing time T . In fact, the existence of
any such equilibrium implies that there are N ! equilibria, by
simply changing the order of arrivals. Nevertheless, we focus
our analysis on symmetric mixed strategy profiles defined by
a single cdf F on the interval [0, T ]. We argue that symmetric
equilibrium is of more interest since it does not require any
coordination or identification of the customers. From now on
we denote the probability of the server being in state i ∈ {0, 1}
at time t by pi (t), although this probability is always a function
of the mixed strategy F being played by all customers. We
further limit the mixed strategies to distributions such that their
support can be defined as a finite union of points and intervals
to avoid technical difficulties (see [10]).
A. General equilibrium properties
Using arguments similar to those used in [1], [3] and [5], we
claim that an equilibrium strategy has the following features:
(1) There is an atom at zero, namely it states some probability pe > 0 with which one arrives at time zero
(pe := F (0)). This is the case since otherwise one’s
best response will always be to try at time zero with
probability one as finding an idle server is then guaranteed.
(2) An interval (0, te ) with te < T in which no one arrives;
F (t) = F (0) for any t ∈ (0, te ).
(3) Some density f (t) > 0 of arriving along the interval
[te , T ] (no other atoms or empty intervals).
We next present the relation between pe and te , and in the
following sections we show how to find f (t), t ∈ [te , T ] for
different assumptions on the population size. It is possible that
in equilibrium pe = 1 and te ≥ T , i.e. all customers arrive at
time zero with probability one.
Consider am arbitrary customer, and suppose all other N
customers arrive at time zero with probability p. Given the
value of N , the number of customers arriving at time zero,
3
denoted by Xp , is clearly binomial . The probability of obtaining service if she decides to arrive at time zero is E Xp1+1 .
The probability of obtaining service by arriving at time t,
assuming all customers arrive with zero probability along the
interval (0, t], is: P(Xp = 0) + P(Xp ≥ 1) (1 − e−µt ). If the
probability to obtain service is equal at times zero and t, then
we get the following equation:
1
= P(Xp = 0) + P(Xp ≥ 1) 1 − e−µt ,
(1)
E
Xp + 1
or equivalently:
e−µt = E
Xp
|Xp ≥ 1 .
Xp + 1
(2)
In equilibrium, pe and te satisfy this condition. Using equation (2) we can always derive te (or pe ) given pe (or te ).
Furthermore, this yields a condition for the equilibrium of all
customers arriving at time zero: if pe = 1 yields te > T then
the only equilibrium is pe = 1.
Lemma 2.1: For any µ, T and N , the time t that satisfies
equation (2) is monotone decreasing with respect to p.
Proof If {Xp |Xp ≥ 1} is stochastically increasing with respect to p then the expected value of the increasing function
Xp
Xp +1 is increasing as well. Thus, the larger is p, the larger is
−µt
e
and hence the smaller is t. It is straightforward to show,
by taking derivative, that P(Xp ≥ x|Xp ≥ 1, N = n) is a
strictly increasing function with respect to p, for any x ≥ 1
and any positive integer n. The unconditional (with respect to
N ) probability is a weighted average of increasing functions
and therefore it is increasing too.
Lemma 2.1 provides a somewhat unintuitive relation between pe and te . In equilibrium, if more customers are
expected to arrive in time zero then the interval with no arrivals
after time zero gets shorter. After some thought, this result
should not be very surprising because a higher value of pe
implies a smaller probability to obtain service at time zero,
and thus at all other times in equilibrium. It is therefore not
unintuitive that the time required for the idle server probability
to increase to this new lower level is shorter.
Remark Throughout this work we will denote the probability
to obtain service at time zero given p, by C(p) := E( Xp1+1 ).
This allows for a convenient statement of the equilibrium
condition that the probability is constant on all of the support.
B. Poisson number of customers
We now assume that the number of customers follows a
Poisson distribution with parameter λ. We characterize the
equilibrium arrival strategy and prove that it is unique.
Theorem 2.2: Consider pe and te that are the unique solution
to the following pair of equations:
1
1 − C(pe )
te = − log
,
(3)
µ
1 − e−λpe
and
1 − pe
µ(1 − C(pe ))
=
,
T − te
λC(pe )
(4)
where C(pe ) = (1 − e−λpe )/λpe . If for pe = 1 the solution of
(3) yields te ≥ T then pe = 1 is the equilibrium. Otherwise
pe < 1 and te < T define the symmetric arrival distribution,
and the uniform equilibrium density on the interval [te , T ] is
f (t) =
µ(1 − C(pe ))
, te ≤ t ≤ T.
λC(pe )
(5)
Proof Let 0 < pe ≤ 1 be the probability of arriving at time
zero in equilibrium. For one who decides to arrive at this
time too, the probability of obtaining service is easily shown
to equal C(pe ) = (1 − e−λpe )/λpe . Using the equilibrium
condition (1) we derive te as given in (3). Denote by p0 (t)
the probability that the server is idle at time t, 0 ≤ t ≤ T . The
equilibrium condition is that p0 (t) = C(pe ) along the interval
[te , T ]. Standard queueing dynamics yield1
p00 (t) = (1 − p0 (t))µ − λf (t)p0 (t), te ≤ t ≤ T.
(6)
The condition that the probability is constant implies that
p00 (t) = 0 for any te ≤ t ≤ T . Hence, by (6) we get the
density in (5). In particular, the arrival density along [te , T ]
is constant. Recall that the probability of arriving during this
interval equals 1 − pe , and since the density is constant we
have f (t)(T − te ) = 1 − pe , leading to the second equilibrium
equation (4). If pe ≥ 1 or equivalently te ≥ T , we conclude
that pe = 1 is the equilibrium arrival strategy. Uniqueness
is obtained by the monotonicity of te with respect to pe ,
established in Lemma 2.1, together with the fact that equation
(4) has a unique solution pe (the lhs expression is decreasing
and the rhs expression is increasing with pe ).
Lemma 2.3: In equilibrium the probability of arriving at time
zero, pe , is monotone decreasing with respect to the closing
time T .
Proof Let pe be the probability to arrive at time zero in an
equilibrium of the arrival game with parameters (λ, µ, T ).
Assume that in the game with parameters (λ, µ, T ∗ ) where
T ∗ > T , there exists an equilibrium such that p∗ ≥ pe is the
equilibrium probability to arrive at time zero in the latter game.
We respectively denote the probabilities to obtain service by
C(pe ) and C(p∗ ), and the ending time of the empty intervals
by te and t∗ . By Theorem 2.2 we have:
1 − pe = (T − te )
(1 − C(pe ))µ
.
λC(pe )
Note that the probability to obtain service is lower when more
customers arrive at time zero: C(pe ) ≥ C(p∗ ) and by Lemma
2.1 we have: te ≥ t∗ . So we can derive:
(1 − C(pe ))µ
1 − pe = (T − te )
λC(pe )
.
(1 − C(p∗ ))µ
∗
< (T ∗ − t∗ )
=
1
−
p
λC(p∗ )
But this is a contradiction of the assumption that pe ≤ p∗ .
Thus, we can conclude that if T ∗ > T then p∗ < pe .
We have shown that a longer service interval means that in
equilibrium, less customers will arrive at time zero and that
1 This is the Kolmogorov backward equation of state 0, in a two state (0
and 1) continuous time Markov chain with rates µ and λf (t).
4
there will be a longer interval with no arrivals subsequently.
Equivalently, the same arguments can be used to show that
increasing the population size λ has the same effect, while
increasing the service rate µ has the opposite effect. In Figure
1, we illustrate the equilibrium arrival cdf for different values
of λ. When the expected population size is bigger, then pe is
smaller and the interval (0, te ) is shorter, i.e. the distribution
is closer to uniform on of all the interval [0, T ].
1
λ = 1
λ = 5
0.5
λ = 20
λ = 100
0
0.5
1
t
Fig. 1. Equilibrium arrival cdf for different values of λ (µ = 2, T = 1).
In table I we see the respective probabilities to obtain service
and the expected number of admitted customers. Unsurprisingly, the number of expected customers who obtain service
approaches the expected number of service completions plus
one, 1 + µ1 (3 in the example), when λ increases. This is
because effectively, there is an arrival at almost every moment
of the interval [0, T ] and a new customer arrives as soon as
the server becomes idle.
TABLE I
E QUILIBRIUM RESULTS FOR DIFFERENT VALUES OF λ (µ = 2, T = 1).
λC(pe )
We now assume that the number of customers is
deterministic and equals N + 1. Note that this is consistent
with the previous definition of N as the number of additional
customers that every customer believes are participating. We
next characterize the symmetric equilibrium strategy and
provide an algorithm for it’s numerical computation.
If the N other customers arrive at time zero with probability p, then for any singled out customer, the probability
of obtaining service if she decides to arrive at time zero
N +1
is C(p) := 1−(1−p)
(N +1)p . Using (1) we obtain the equation
relating pe and te :
F (t)
C(pe )
C. Deterministic number of customers
λ=1
λ=5
λ = 20
λ = 100
0.763
0.381
0.131
0.029
0.763
1.907
2.615
1 − (1 − pe )N +1
= (1 − pe )N + (1 − (1 − pe )N ) 1 − e−µte .
(N + 1)pe
(7)
We seek a density function such that if all N other customers use this strategy too, any given customer has a constant
probability to find an idle server at any point in the interval
[te , T ]. The arrival process is not Markovian, so instead we
consider the process with states (i, j) where i ∈ {0, 1}
indicates if the system is idle or not, and j ∈ {0, ..., N } is the
number of arrivals so far. This is indeed a Markov process (not
homogeneous in time). We further denote the hazard function
of the arrival distribution at time t by h(t) := f (t)/(1−F (t)).
If we denote the arrival rate at time t when j customers have
f (t)
already arrived by λj (t), then λj (t) = (N − j) 1−F
(t) (this
is the sum of hazard rates for the remaining customers). The
dynamics of this process are illustrated in Figure 2.
Arrivals
0 ...
j
j+1
... N
Idle
...
0,I j
0, jI + 1
...
µ
Busy
2.911
...
λj (t)
1, j
µ
.
λj (t)
1, j + 1
...
Fig. 2. Arrival and service process diagram.
Remark The equilibrium strategy found here resembles the
one derived in [6]. The model stated there is the same loss
system defined here and at time t = 0 the server is idle.
Customers arrive in accordance with a Poisson process and
their time of arrival is their private information. Based on this
information they need to decide if to try or not to try. A trial
costs K and the reward due to service completion is R. To
avoid trivialities, it is assumed that R > K. Here one looks for
an equilibrium try or not try symmetric strategy. This strategy
will state a trial probability q(t) for any time t, t ≥ 0. It was
proved in [6] that,



 1 0 ≤ t < − 1 log 1−qe
λ+µ
λ+µqe
,
q(t) =


1−qe
1
 qe
t ≥ − λ+µ
log λ+µq
e
where qe = µλ R−K
K . In particular, q(t), t ≥ 0, is a two-level
step function.
The probability of state (i, j) at time t, denoted by pi,j (t),
satisfies the backward equations:
0
p0,j (t) = µp1,j (t) − λj (t)p0,j (t), te ≤ t ≤ T, 0 ≤ j ≤ N
0
p1,j (t) = λj−1 (t)(p0,j−1 (t) + p1,j−1 (t)) − (µ + λj (t))p1,j (t),
te ≤ t ≤ T, 1 ≤ j ≤ N .
(8)
The probability of obtaining service, when arriving at time
t, is the sum of the joint probabilities
j arrivals until time
Pof
N
t and the server being idle: p0 (t) = j=0 p0,j (t). Recall that
in equilibrium this probability is constant and equal to C(pe )
along the support, therefore the sum of the derivatives given
by (8) equals zero:
0=µ
N
X
j=1
p1,j (t) − h(t)
N
X
j=0
(N − j)p0,j (t).
5
The first sum is the probability that at time t the server is
busy, and so it equals 1 − C(pe ) for any t ∈ [te , T ]. Therefore
an equilibrium condition is
h(t)
N
X
(N − j)p0,j (t) = µ(1 − C(pe )).
(9)
j=0
The initial conditions for the probabilities at time te are:
p0,j (te ) = b(j; N, pe ) 1 − 1{j > 0}e−µte , 0 ≤ j ≤ N,
(10)
and
p1,j (te ) = b(j; N, pe )1{j > 0}e−µte , 0 ≤ j ≤ N,
(11)
where b(j; N, p) := Nj pj (1 − p)N −j .
We summarize the above analysis in the following theorem.
Theorem 2.4: Any symmetric equilibrium arrival distribution with cdf F satisfies the following properties:
1) There exists a positive probability to arrive at time zero,
pe := F (0).
2) The probability that an individual customer obtains
N +1
e)
.
service is C(pe ) := 1−(1−p
(N +1)pe
3) There are no arrivals in the interval (0, te ), where
N pe − (1 − (1 − pe )N )
1
.
(12)
te = − log
µ
N pe (1 − (1 − pe )N −1 )
4) If for pe = 1 the solution to (12) yields te ≥ T then
pe = 1, i.e. all customers arrive at time zero.
5) Otherwise, pe < 1 and the continuous arrival distribution
is defined using the functional differential equation:
µ(1 − C(pe ))
h(t) = PN
, t ∈ [te , T ],
j=0 (N − j)p0,j (t)
(13)
f (t)
where h(t) := 1−F
(t) is the hazard rate of the arrival
distribution and pi,j (t) are the resulting probabilities of
the process defined above.
RT
6) The equilibrium density satisfies 1 − pe = te f (t)dt.
Proof Properties 1 and 2 are special cases of the general equilibrium properties presented in Section II-A. The value of te
given pe in (12) is a direct result of (7), which leads to property
3. Property 4 is a result of the monotone relation between pe
and te established in Lemma 2.1. Finally, properties 5 and
6 summarize the analysis preceding to the statement of the
theorem.
Lemma 2.5: For any t ∈ [0, T ], the cdf of the equilibrium
arrival distribution, F (t), characterized in Theorem 2.2 is
increasing with respect to the initial condition pe . Furthermore,
the equilibrium density can be stated as:
f (t) =
µ
P(BN (t) = 1)
·
,
N P(BN −1 (t) = 0)
(14)
where Bk (t) ∈ {0, 1} is the busy period process of the server
(1 when busy and 0 when idle), when there are k ∈ {1, . . . , N }
customers arriving according to the equilibrium F .
Remark It is important to note that the auxiliary process
Bk (t) is defined using the same equilibrium arrival distribution
F of the N customer game and not that of a k customer game.
The probabilities can be stated using the two state
PN process
defined before, for example P(BN (t) = i) =
j=0 pi,j (t)
for i ∈ {0, 1}. For the exact representation of the probability
P(Bk (t) = i), a slight modification of notation is required and
we leave these details for the appendix, along with the proof
of Lemma 2.5 which is rather long and technical.
Corollary 2.6: There exists a unique symmetric equilibrium
arrival distribution.
The corollaryRis derived directly from the equilibrium condiT
tion 1−pe = te f (t)dt, since the lhs is decreasing and the rhs
is increasing, w.r.t. pe . On top of the uniqueness result, Lemma
2.5 also yields a technical advantage which will be used in
the numerical procedure we establish in the next section: The
hazard rate of the equilibrium arrival distribution in equation
(13) is unbounded and approaches infinity when t → T .
This can cause a numerical approximation of the equilibrium
F (t) which relies on discrete increments using the differential
dynamics of (8) to be inaccurate for values t that are close to
T . The density function in (14) is however bounded and well
defined even at time T .
Example: N = 1 (two customer game): If there is only one
other customer then clearly P(BN −1 (t) = 0) = 1, as there
are no arrivals at all. Hence, from (14) we can conclude that
the equilibrium arrival distribution is uniform on the interval
[te , T ]:
µpe
P(B1 (t) = 1)
= µ(1 − C(pe )) =
.
P(B0 (t) = 0)
2
RT
And by solving 1 − pe = te f (t)dt we can derive the
2
equilibrium values te = logµ 2 and pe = 2+T µ−log
2 , and the
1+T µ−log 2
probability to obtain service C(pe ) = 2+T µ−log 2 .
f (t) = µ
D. Numerical Procedure
We now describe a numerical procedure to compute the
equilibrium arrival distribution. First compute te for pe = 1,
if te ≥ T then all arriving at time zero is the equilibrium. Otherwise, use the following algorithm with precision parameters
∆ > 0 and > 0:
(1) Pick an arbitrary pe ∈ (0, 1)
(2) Compute te , p0,j (te ) and p1,j (te ), for j = 0, . . . , N ,
using (12), (10) and (11), respectively.
(3) Compute pi,j (t + ∆) and f (t + ∆), for i ∈ {0, 1} and
j = 0, . . . , N , in small increments of ∆:
pi,j (t + ∆) = pi,j (t + ∆) + ∆p0i,j (t + ∆),
using (8) and (14), respectively. Stop when reaching t∗
such that |F (t∗ ) − 1| ≤ .
(4) If |t∗ − T | ≤ stop. Else, if t∗ < T then pick a smaller
pe and if t∗ > T then pick a larger pe , and go back to
(2).
Remark A bisection based method can be applied for finding
the equilibrium pe . Lemma 2.5 states that the equilibrium
arrival distribution F is stochastically increasing with respect
to pe , hence the bisection method converges to the unique
solution.
6
E. Numerical examples
We computed the equilibrium arrival distribution for different population sizes and service rates, and display the results
in Figure 3 and Table II. We have shown that the arrival
distribution is uniform if N = 2, as in the Poisson case.
For N > 2 the density is not uniform but appears to be
”close” to uniform, especially for larger values of N where the
distribution becomes spread out on almost all of the interval
[0, T ].
f (t)
III. S OCIAL OPTIMIZATION
We define the social utility as the expected sum of all the
customers’ utility, and in this case the utility of a customer is
the probability to obtain service. Therefore, the overall social
utility is the expected number of customers obtaining service.
We have characterized the equilibrium arrival strategies, and
the numerical analysis of section II-D can also be applied to
compute the social utility. We now seek a socially optimal
schedule of arrivals for a deterministic number of customers
N , and subsequently the price of anarchy which we define as:
P oA(N ) :=
0.6
N = 2
0.4
N = 4
N = 10
0.2
N = 20
0
0.2
0.4
0.6
0.8
t
1
∗
UN
e ,
UN
(15)
∗
e
where UN
is the optimal social utility and UN
is the social
utility of the symmetric mixed strategy equilibrium2 . The
definition of the socially optimal utility is dependent on the
degree of control that is given to the central planner. If the
central planner observes service completions and immediately
sends a new customer then the socially optimal utility is
∗ := E max{Y
UN
(µT ) + 1, N },
Fig. 3. Arrival strategy for different values of N (µ = T = 1).
TABLE II
E QUILIBRIUM RESULTS FOR DIFFERENT VALUES OF N (µ = T = 1).
N =2
N =4
N = 10
N = 20
pe
0.867
0.694
0.589
0.547
C(pe )
0.567
0.357
0.170
0.091
N C(pe )
1.133
1.429
1.698
1.829
F. Any distribution of the number of customers
0
Suppose that the total number of customersP
N is a discrete
∞
random variable with a pmf π(n) such that n=0 π(n) = 1,
and mean value of n̄. Note that if a customer is chosen to
arrive the posterior distribution of the additional customers
arriving is given by γ(n) = (n+1)π(n+1)
, n ≥ 0, we denote
n̄
this random variable by N . See [13] and [7] for further details
on the properties of games with random number of players.
The analysis of the previous section can be applied to
the case where N follows any probability distribution on the
positive integers. This generalization is described in detail in
[10]. In the context of our model, the dynamics of the process
f (t)
we will now have λj (t) = E(N − j|N ≥ j) 1−F
(t) as the
non-homogeneous arrival rate. For a distribution with infinite
support there will be some approximation needed, but this
should not be a problem since the probabilities of large n
converge to zero. The dynamics of the process given by (8),
with the new arrival rate. If Xpe is the number of arrivals at
time zero, then
k
∞
P(Xpe = j) = Σk=j P(N = k)
pe j (1 − pe )k−j .
j
The initial conditions in (10) and (11) are updated accordingly.
where Y(µT ) ∼ P oisson(µT ). Note that for large values of
∗ ≈ µT + 1.
N , we have UN
If the central planner cannot dynamically assign arrival
times given the realization of the process, but rather has to
choose a predetermined schedule, then the socially optimal
∗ . When considering predeterutility is upper bounded by UN
mined schedules, we have two options: the central planner
can assign individual (non-symmetric) arrival times to all
customers, or the central planner can only assign a single
(symmetric) strategy to all customers. The socially optimal
utility of the second is clearly upper bounded by that of the
first, and both are upper bounded by the dynamic optimal
∗ . In the remainder of this section we provide
solution, UN
an explicit solution for the non-symmetric case and leave the
symmetric schedule analysis to future work.
A. Non symmetric strategies
We first rule out randomized policies using the fact that
any realization can be achieved by a deterministic policy. The
optimal deterministic policy may not be unique but there is
no obligation to randomize in order to achieve the socially
optimal utility. We also rule out any arrival policy that does
not schedule an arrival time zero and an arrival at time T ,
as any such policy can be trivially improved by moving the
first (or last) arrival to 0 (or T ). Any deterministic arrival
policy starting at time zero can be defined by the ordered time
intervals between each two arrivals. We denote these interarrival times by t1 , . . . , tN , where the customers are ordered
arbitrarily. We denote the set of possible arrival schedules by
(
)
N
X
T := (t1 , ..., tN ) : t1 = 0, ti ≥ 0 ∀i ≥ 2,
ti = T .
i=1
Theorem 3.1: If there is a deterministic number of customers
N > 1 seeking service from a single server, the socially
2 We again focus on the symmetric equilibrium, even though there may be
multiple pure strategy equilibria with possibly lower social utility
7
optimal strategy profile is for the customers to arrive in fixed
time intervals of length NT−1 . The socially optimal utility is
then equal to:
µT
∗
UN
= N − (N − 1)e− N −1 .
(16)
Proof For any arrival schedule t ∈ T , the probability of
customer 1 obtaining service is of course one. The probability
of customer 2 obtaining service is the probability of customer 1
completing service by time t2 , and of any customer 1 < j ≤ k
it is the probability of the previous customer completing
service within tj units of time, which is the period since the
last arrival. Note that it makes no difference to customer j
which customers obtained service before her. The expected
number of served customers when applying schedule t is
UN (t) = 1 +
N
X
(1 − e−µti ) = N −
i=2
N
X
e−µti .
proportion of the customers. However, when the population is
large enough then both the equilibrium and optimal schedules
again yield similar results, which is expected as they are both
sending customers at virtually every moment and thus almost
fully utilizing the server capacity.
10
e
UN
5
1
∗
UN
µT + 1
2
20
40
60
80
N
100
Fig. 4. Expected number of customers served in equilibrium and under the
socially optimal schedule, for different values of N (µ = 10, T = 1).
i=2
The optimization problem can then be stated as a convex
program:
min
t∈T
n
X
P oA(N )
1.4
e−µti
i=1
(
s.t. ti ≥ 0, ∀i ∈ {1, . . . , n},
n
X
(17)
)
ti = T
1.3
1.2
,
i=1
1.1
where n = N −1. Indeed, the solution to (17) is ti = n1 , ∀i =
1, . . . , n.3 Furthermore, the social utility is maximized by t1 =
0 and tj = NT−1 for any j = 2, . . . , N . Finally, the socially
optimal expected number of served customers is given by (16).
1
2
20
40
60
80
N
100
Fig. 5. Price of anarchy for different values of N (µ = 10, T = 1).
Corollary 3.2: The price of anarchy for a deterministic
number of customers and a single server is
P oA(N ) =
N − (N − 1)e
− NµT
−1
1−(1−pe )N
pe
,
(18)
where pe is the equilibrium probability to arrive at time zero
as characterized in Section II-C.
B. Numerical examples
In Figure 4 we compared the expected number of customers
served in equilibrium to that of the socially optimal schedule
for population sizes N = 2, . . . , 100, when the service rate is
µ = 10 and the closing time is T = 1. When the population
is large then the number of served customers approaches the
maximum service capacity, µT + 1, for both the equilibrium
and the optimal schedule. When the population size is small,
compared to the service rate, then the equilibrium social utility
is close to that of the optimal schedule. This is also seen
in Figure 5 where the Price of Anarchy is computed for the
same values. This non-monotone behaviour implies that if the
number of competing customers is small then the equilibrium
behaviour spreads them out on the interval efficiently, but
when it is larger then the competition leads to a loss of a bigger
3 The objective function is a sum of identical convex and strictly decreasing
functions.
IV. C ONCLUSION AND FUTURE WORK
We have presented here a single server loss system arrival
game and provided analysis of the Nash equilibrium behaviour
and of the social optimization problem. In particular, we have
shown that the equilibrium arrival distribution has an atom
at zero and a continuous density on a subset of the service
interval. A uniform density is derived for the case in which
the number of customers follows a Poisson distribution. A
general characterization of the equilibrium is provided for any
deterministic number of customers, along with a numerical
procedure to compute it. We have also presented an explicit
solution for the social optimization problem when the central
planner can choose a predetermined arrival schedule. The price
of anarchy is shown to be non-monotone with respect to the
population size, which implies that the equilibrium arrival
distribution is efficient for small and large populations but not
so much when the population is close to the server capacity.
A possible extension to our model is to consider a system
with multiple servers and a buffer with limited capacity.
Another possible avenue to explore is to approximate the
model for large populations, using a fluid model. The social
optimization analysis carried out here assumed that the central
planner can dictate individual arrival times to all participating
customers. This may not be feasible when there is a large
or even random population of customers. Therefore, finding a
8
socially optimal symmetric strategy is of much interest. In [3]
a numerical discretization method is applied to finding such
an optimal symmetric strategy for a single server queue with
a closing time and waiting costs. They find that the optimal
strategy, which is unsurprisingly randomized, is approximately
a uniform distribution on the arrival interval. This method
may also be applied for approximating the optimal symmetric
schedule for the model presented here.
A PPENDIX
Proof of Lemma 2.5 We will show that the equilibrium cdf,
F (t), is an increasing function of the equilibrium probability
to arrive at time zero, pe for any t ∈ [0, T ]. This is clearly
true for t ∈ [0, te ], as there are no arrivals in this interval and
therefore F (t) = pe . In the remainder of this proof we show
that the equilibrium density function, f (t) is also an increasing
function of pe for any t ∈ [te , T ).
We denote the equilibrium arrival time of customer i ∈
{1, . . . , N } by Ti ∼ F . We further denote the number of
arrivals until and the state of the server (busy=1 or idle=0) at
time t by A(t) and B(t), respectively. The hazard rate of the
equilibrium arrival distribution in (13) can also be stated as
µP(B(t) = 1)
f (t)
=
,
1 − F (t)
N P(B(t) = 0) − ϕ(t)
where
ϕ(t):=
N
X
jp0,j (t) = EA(t)1{B(t)=0}
j=0


.
N
X
= E
1{B(t)=0,Tj ≤t}  = N E1{B(t)=0,T1 ≤t}
j=1
If t → T then E1{B(t)=0,T1 ≤t} → P(B(t) = 0), and
the hazard rate approaches infinity, since the numerator is
constant. Note that is the case for any distribution with a finite
support. Otherwise, for any t < T
ϕ(t) = N P(B(t) = 0, T1 ≤ t)
= N (P(B(t) = 0) − P(B(t) = 0, T1 > t))
,
= N (P(B(t) = 0) − P(B(t) = 0|T1 > t)(1 − F (t)))
hence the density for t ∈ [te , T ) is
f (t) =
µ
P(B(t) = 1)
·
.
N P(B(t) = 0|T1 > t)
(19)
The probability P(B(t) = 0|Tj > t) is in fact the probability
of an idle server at time t, when there are only N −1 customers
that arrive according to F . This is because any arrival time is
independent of any other arrival or service time. If we denote
the busy period process with k customers by Bk (t), then the
equilibrium density is given by:
f (t) =
P(BN (t) = 1)
µ
·
.
N P(BN −1 (t) = 0)
(20)
The expression on the rhs is well defined for t = T , and
because the density is bounded and all terms are continuous
we have that
P(BN (t) = 1)
P(BN (T ) = 1)
lim
=
.
t→T P(BN (t) = 0|T1 > t)
P(BN −1 (T ) = 0)
If pe increases, then P(BN (t) = 1) = 1 − C(pe ) increases
for all t ∈ [te , T ]. We next argue that if pe is increases then
P(BN −1 (te ) = 0) decreases. If Xp ∼ Bin(N − 1, pe ), then:
P(BN −1 (te ) = 0) = 1 − P(BN −1 (te ) = 1)
= 1 − P(Xp ≥ 1)e−µte .
Since both terms in the rhs product are increasing with pe (see
Lemma 2.1), we conclude that P(BN −1 (te ) = 0) is decreasing
and f (te ) and h(te ) are both increasing. We can now carry
the argument forward because the service rate, µ, is constant
and the arrival rate determined by h(t) is higher. Finally, if
this is the case then for any t > te , f (t) in (20) is increasing
w.r.t pe and so is F (t).
Acknowledgment: The authors gratefully acknowledge the
financial support of the Israel Science Foundation grant no.
1319/11.
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