Equilibrium customer strategies in a single server Markovian queue ...

Equilibrium customer strategies in a single server Markovian queue with setup times Apostolos Burnetas and Antonis Economou {aburnetas,aeconom}@math.uoa.gr Department of Mathematics, University of Athens Panepistemioupolis, Athens 15784, Greece

Published in 2007 in Queueing Systems 56, 213-228 The original publication is available at www.springerlink.com DOI 10.1007/s11134-007-9036-7

Abstract. We consider a single server Markovian queue with setup times. Whenever this system becomes empty, the server is turned off. Whenever a customer arrives to an empty system, the server begins an exponential setup time to start service again. We assume that arriving customers decide whether to enter the system or balk based on a natural reward-cost structure, which incorporates their desire for service as well as their unwillingness to wait. We examine customer behavior under various levels of information regarding the system state. Specifically, before making the decision, a customer may or may not know the state of the server and/or the number of present customers. We derive equilibrium strategies for the customers under the various levels of information and analyze the stationary behavior of the system under these strategies. We also illustrate further effects of the information level on the equilibrium behavior via numerical experiments. Keywords: queueing, setup times, vacations, balking, continuous time Markov chain, equilibrium strategies, individual optimization, pricing, stationary distribution, difference equations, matrix analytic methods 1. Introduction Queues with removable servers (also referred to as queueing systems with vacations) deal with situations where the servers may be unavailable for serving customers over some intervals of time. Such situations are often incurred in real applications. For example, a server may be deactivated for economic reasons (low traffic intensity and/or high stand-by costs), suffer random failures, go under preventive maintenance or attend to a secondary system. Due to their versatility and applicability, queueing systems with removable servers have been extensively studied. Detailed surveys are contained in Takagi (1991) and Tian and Zhang (2006). Although the literature for the single server case is very rich, for the multiserver case the number of published papers is limited. A classification of most of them is presented in Artalejo and Lopez-Herrero (2003). A significant part of this literature is dedicated to the analysis of queues with setup times. In such models once a server is reactivated, a generally random time is required for setup before it can begin serving customers. Bischof (2001), Choudhury (1998, 2000), He and Jewkes (1995) and the references therein consider various single server systems with setup times, while Borthakur and Choudhury (1999) and Artalejo et al. (2005) deal with some multiserver models. In particular, the performance evaluation for an M/M/1 queue with server setups has been 1

carried in Adan and van der Wal (1998),p.51-54 and its multiserver counterpart in Artalejo et al. (2005). In the queueing literature there is also an emerging tendency to study systems from an economic viewpoint. More concretely, it is assumed that there exists a reward-cost structure for the customers of a given system, incorporating their desire for service as well as their dislike for waiting. The customers are allowed to make decisions as to whether to join or balk, buy priority or not etc. This can be viewed as a game among the customers. The basic problem is to find individual and social optimal strategies. These ideas go back at least to the pioneering works of Naor (1969) and Edelson and Hildebrand (1975) who studied equilibrium and socially optimal strategies for whether to join or balk in an M/M/1 queue with a simple reward-cost structure. Naor (1969) studied the case where each customer observes the queue length before his decision, while Edelson and Hildebrand (1975) considered the unobservable case. These results were further refined and extended by several authors, see e.g. Yechiali (1971), Hassin (1986) and Chen and Frank (2001, 2004). Moreover, several authors have investigated the same problem for various queueing systems incorporating many diverse characteristics such as priorities, reneging and jockeying, schedules and retrials etc. The fundamental results in this area with extensive bibliographical references can be found in the comprehensive monograph of Hassin and Haviv (2003). In the present paper we investigate the equilibrium customer behavior in a single server Markovian queue with setup times. The customers’ dilemma is whether to join the system or balk. We examine various cases with respect to the level of information available to customers before they make this decision. More specifically, at his arrival epoch a customer may or may not know the state of the server and/or the number of customers present. Therefore, four combinations emerge, ranging from no to full information. In each of the four cases we characterize customer equilibrium strategies, analyze the stationary behavior of the corresponding system and derive the social benefit for all customers. We also explore the effect of the information level on the equilibrium behavior and the social benefit via analytical and numerical comparisons. When no information is available to arriving customers, the stationary analysis turns out to be equivalent to that of the M/M/1 system with server setups and no customer decisions, performed in Adan and van der Wal (1998). However, in all other cases, to derive the stationary behavior this analysis must be generalized in different directions. To our knowledge, these generalizations are new. The paper is organized as follows. In Section 2, we describe the dynamics of the model, the reward-cost structure and the decision assumptions for the customers (information levels). In Section 3 we consider the case where the customers know the number of customers in system before they decide whether to join or to balk. We distinguish two subcases depending on the additional information, or lack thereof, of the server state. We determine equilibrium threshold strategies and investigate the resulting stationary system behavior. In Section 4 we consider the unobservable case. We also consider the same two subcases, derive the corresponding mixed equilibrium strategies, and analyze the stationary behavior. Finally, in Section 5, we present several numerical experiments that demonstrate the effect of the information level to the various performance measures. 2. Model description We consider a single server queueing system with infinite waiting room in which customers arrive according to a Poisson process with rate λ. The service times of the customers are assumed to be exponentially distributed random variables with rate µ. The server is deactivated as soon as the queue becomes empty. When a new customer arrives at an empty system, a setup process 2

(1, 1) o O

ww wwµ θ w w {ww λ / (0, 0) (1, 0)

λ µ

/

(2, 1) o O

λ

/

··· o

µ

λ

/

(n, 1) o O

µ

θ λ

λ µ

θ

/ (2, 0) λ

/ ···

λ

/ (n, 0)

/

(n + 1, 1) o O

λ µ

/

···

θ λ

/ (n + 1, 0)

λ

/ ···

Figure 1: Transition rate diagram of the original model starts for the server to be reactivated. The time required for setup is also exponentially distributed with rate θ. During the setup customers continue to arrive. We assume that interarrival times, service times and setup times are mutually independent. We represent the state at time t by the pair (N (t), I(t)), where N (t) denotes the number of customers in the system and I(t) denotes the state of the server. It is clear that the process {(N (t), I(t)) : t ≥ 0} is a continuous time Markov chain with state space S = {(n, i) | 0 ≤ i ≤ 1, n ≥ i }. The transition diagram is shown in Figure 1. In Figure 1 we observe that starting from a state (n, 1), n ≥ 1 it is not possible to make transitions to states (m, 0) without visiting state (0, 0) first. Due to this special structure, it is possible to compute explicitly the stationary distribution and several performance measures under various customer strategies. We are interested in the behavior of customers when they can decide whether to join or balk upon their arrival. To model the decision process, we assume that every customer receives a reward of R units for completing service. This may reflect his satisfaction and/or the added value of being served. On the other hand there exists a waiting cost of C units per time unit that the customer remains in the system (in queue or in service). Customers are risk neutral and maximize their expected net benefit. We assume that R>

C C + . µ θ

(1)

This condition ensures that the reward for service exceeds the expected cost for a customer who finds the system empty. Otherwise, after the system becomes empty for the first time no customers will ever enter. Finally, the decisions are irrevocable: retrials of balking customers and reneging of entering customers are not allowed. Under the above framework we can think of the situation as a symmetric game among the customers since they are all indistinguishable. Denote the common set of strategies and the payoff function by S and F , respectively. More concretely let F (a, b) be the payoff of a customer that selects strategy a when everyone else selects strategy b. A strategy se is a (symmetric Nash) equilibrium if F (se , se ) ≥ F (s, se ), for every s ∈ S. The intuition is that an equilibrium strategy is a best response against itself, i.e., if all customers agree to follow it no one can benefit by changing it. A strategy s1 is said to dominate strategy s2 if F (s1 , s) ≥ F (s2 , s), for every s ∈ S and for at least one s the inequality is strict. A strategy s∗ is said to be weakly dominant if it dominates all other strategies in S. The notion of dominance is substantially stronger than that of equilibrium. In fact, while an equilibrium strategy exists in almost all situations, a weakly dominant strategy rarely does. In the next two sections we obtain equilibrium customer strategies for joining/balking. We distinguish four cases depending on the information available to customers at their arrival instant, before the decision is taken: • Fully observable case: Customers observe (N (t), I(t)) 3

• Almost observable case: Customers observe only N (t) • Almost unobservable case: Customers observe only I(t) • Fully unobservable case: Customer do not observe the system state For the terminology, note that the information N (t) = n, for n ≥ 1, implies that the system is in one of the two states {(n, 0) and (n, 1)} (or the single state {(0, 0)}, for n = 0}). On the other hand, the information I(t) = i, for i = 0, 1, implies that the state belongs to the infinite set {(n, i), (n + 1, i), ...}. It is for this reason that we refer to these cases as ‘almost observable’ and ‘almost unobservable’, respectively. It must be emphasized that by employing this terminology, we don’t imply that it is more critical to have information about N (t) than about I(t). This is not true in general. For example in a system with very low setup rate and very high service rate it is intuitively expected that observing I(t) is more important than observing N (t). We further discuss the value of information under the light of the numerical results in Section 5. From a methodological point of view, the observable cases are similar and are treated in parallel in Section 3. The unobservable cases are both treated in Section 4. 3. Equilibrium threshold strategies for the observable cases For the fully observable and almost observable cases we show that there exist equilibrium strategies of threshold type. Specifically there exist threshold levels such that an arriving customer enters the system if the number of customers present upon arrival does not exceed the specified thresholds. We begin with the fully observable case in which customers know the exact state of the system (n, i) upon arrival. A customer who joins the system when he observes state (n, i) has 1−i mean sojourn time equal to n+1 µ + θ , thus his expected net benefit is R−

C(n + 1) C(1 − i) − . µ θ

Such a customer strictly prefers to enter if this value is positive and is indifferent between entering and balking if it equals zero. Assuming that customers break ties in favor of entering, µ(1−i) the tagged customer enters if and only if n + 1 ≤ Rµ . We thus conclude the following. C − θ Theorem 1 In the fully observable M/M/1 queue with setup times there exist thresholds µ¹ º ¹ º ¶ Rµ µ Rµ (ne (0), ne (1)) = − − 1, −1 , C θ C

(2)

such that the strategy ‘observe (N (t), I(t)), enter if N (t) ≤ ne (I(t)) and balk otherwise’ is a unique equilibrium in the class of threshold strategies. Moreover, it is also a weakly dominant strategy. For the stationary analysis, note that if all customers follow the threshold strategy in (2) the system follows a Markov chain similar to that in Figure 1, with state space restricted to Sf o = {(n, 0) | 0 ≤ n ≤ ne (0) + 1} ∪ {(n, 1) | 1 ≤ n ≤ ne (1) + 1} and identical transition rates. The transition diagram is depicted in Figure 2.

4

(1, 1) o O

ww wwµ θ w w {ww λ / (0, 0) (1, 0)

λ µ

/

(2, 1) · · · (ne (0), 1) o O

O

θ λ

λ

/

(ne (0) + 1, 1) · · · (ne (1), 1) o O

µ

θ

λ µ

/

(ne (1) + 1, 1)

θ

/ (2, 0) · · · (ne (0), 0) λ

/ (ne (0) + 1, 0)

Figure 2: Transition rate diagram for the (ne (0), ne (1)) threshold strategy The corresponding stationary distibution (pf o (n, i) : (n, i) ∈ Sf o ) is obtained as the unique positive normalized solution of the following system of balance equations. p(0, 0)λ = p(1, 1)µ

(3)

p(n, 0)(λ + θ) = p(n − 1, 0)λ, n = 1, 2, . . . , ne (0) p(ne (0) + 1, 0)θ = p(ne (0), 0)λ

(4) (5)

p(1, 1)(λ + µ) = p(1, 0)θ + p(2, 1)µ

(6)

p(n, 1)(λ + µ) = p(n − 1, 1)λ + p(n, 0)θ + p(n + 1, 1)µ, n = 2, 3, . . . , ne (0) p(ne (0) + 1, 1)µ = p(ne (0), 1)λ + p(ne (0) + 1, 0)θ p(n, 1)µ = p(n − 1, 1)λ, n = ne (0) + 2, . . . , ne (1) + 1 Define ρ=

(7) (8) (9)

λ λ , σ= . µ λ+θ

By iterating (4) and (9), taking into account (3) and (5), we obtain µ n σ p(1, 1), n = 0, 1, . . . , ne (0) λ µ ne (0) p(ne (0) + 1, 0) = σ p(1, 1) θ p(n, 1) = ρn−ne (0)−1 p(ne (0) + 1, 1), n = ne (0) + 2, . . . , ne (1) + 1. p(n, 0) =

(10) (11) (12)

From (7) it follows that (p(n, 1) : n = 1, 2, ..., ne (0) + 1) is a solution of the nonhomogeneous linear difference equation with constant coefficients µxn+1 − (λ + µ)xn + λxn−1 = −θp(n, 0) = −

θµ n σ p(1, 1), n = 2, 3, . . . , ne (0), λ

(13)

where the last equation is due to (10). Using the standard approach for solving such equations, see e.g. Elaydi (1999),p.70-75 we consider the corresponding characteristic equation µx2 − (λ + µ)x + λ = 0, which has two roots at 1 and ρ. Then the general solution of the homogeneous version of (13) = A 1n + B n 1n (if ρ = 1) . The general solution = A 1n + B ρn (if ρ 6= 1) or xhom is xhom n n gen gen spec xn of (13) is given as xn = xhom + xn , where xspec is a specific solution of (13). Because the n n n p(1, 1) of (13) is geometric with parameter σ, we can again use the nonhomogeneous part − θµ σ λ standard approach to find a specific solution. To this end, we consider specific solutions of the form Cσ n (if σ 6= 1, ρ), Cnσ n (if (σ = 1 or σ = ρ) and ρ 6= 1) or Cn2 σ n (if σ = 1 = ρ). For brevity in the exposition, we assume hereafter that σ 6= ρ and both are different from 1, which 5

is the regular case. The other cases can be viewed as singular and the corresponding results can easily be derived by taking appropriate limits of the regular case results (see Remark 4 below). For the regular case, substituting xn = Cσ n in (13) we obtain C=

µ(λ + θ) p(1, 1). λ(µ − λ − θ)

(14)

Hence, the general solution of (13) is given as n n n xgen n = A 1 + B ρ + C σ , n = 1, 2, . . . , ne (0) + 1

(15)

where C is given by (14) and A, B are to be determined. From (15), for n = 1 it follows that A + ρB = −

λ+θ p(1, 1). µ−λ−θ

(16)

Furthermore, substituting (15) in (6) it follows after some rather tedious algebra that A + ρ2 B = −

(λ + θ)ρ p(1, 1). µ−λ−θ

(17)

(λ+θ)µ Solving the system of (16) and (17) we obtain A = 0 and B = − (µ−λ−θ)λ p(1, 1), thus, from (15), µ(λ + θ) (σ n − ρn )p(1, 1), n = 1, 2, . . . , ne (0) + 1. (18) p(n, 1) = λ(µ − λ − θ)

We have thus expressed all stationary probabilities in terms of p(1, 1), in relations see (10)(12) and (18). The remaining probability, p(1, 1), can be found from the normalization equation ne (0)+1

X

n=0

ne (1)+1

p(n, 0) +

X

p(n, 1) = 1.

n=1

After some algebraic simplifications, we can reduce the stationary probabilities in terms of ρ and σ and summarize the results in the following. Proposition 1 Consider an M/M/1 queue with setup times and σ 6= 1 6= ρ 6= σ, in which the customers follow the threshold policy (ne (0), ne (1)). The stationary probabilities (pf o (n, i) : (n, i) ∈ Sf o ) are as follows · 1 1 − σ ne (0)+1 pf o (1, 1) = ρ(1 − σ)(1 − ρ) (1 − σ)(1 − ρ) Ã µ ¶ne (0)+1 !#−1 σ 1 ρne (1)+2 1 − (19) + (1 − ρ)(σ − ρ) ρ pf o (n, 0) = pf o (ne (0) + 1, 0) = pf o (n, 1) = pf o (n, 1) =

1 n σ pf o (1, 1), n = 0, 1, . . . , ne (0) ρ 1 σ ne (0)+1 pf o (1, 1) ρ(1 − σ) 1 (σ n − ρn )pf o (1, 1), n = 1, 2, . . . , ne (0) σ−ρ 1 (σ ne (0)+1 − ρne (0)+1 )ρn−ne (0)−1 pf o (1, 1), σ−ρ n = ne (0) + 1, . . . , ne (1) + 1. 6

(20) (21) (22)

(23)

Because of the PASTA property, the probability that an arrival finds the system at state (ne (0) + 1, 0) or (ne (1) + 1, 1) and, therefore, balks, is equal to pf o (ne (0) + 1, 0) + pf o (ne (1) + 1, 1). Hence the social benefit per time unit when all customers follow the threshold policy (ne (0), ne (1)) equals Sf o = Rλ(1 − pf o (ne (0) + 1, 0) − pf o (ne (1) + 1, 1))   ne (1)+1 ne (0)+1 X X npf o (n, 0) + npf o (n, 1) . −C  n=0

(24)

n=1

Remark 1. Proposition 1 holds for the stationary distribution corresponding to any threshold policy (ne (0), ne (1)) and not only to the individually optimal policy specified by (2). Remark 2. There is no need to solve from scratch the difference equations for the singular cases, when the conditions σ 6= 1 6= ρ 6= σ do not all hold. The stationary probabilities in the cases can be found by considering limits in the formulae of Proposition 2. For example, assume that σ = ρ and we want to compute pf o (n, 1), for n = 1, 2, . . . , ne (0). We can take limits in (22) as σ → ρ and obtain pf o (n, 1) = nρn−1 pf o (1, 1), n = 1, 2, . . . , ne (0). Remark 3. Equation (24) can be considerably simplified, as the related sums involve only geometric series terms and can be expressed in closed form. We do not elaborate further on this point, as we only use Sf o in numerical experiments. We next consider the almost observable case, where arriving customers only observe the number of customers in the system. Then, the mean sojourn time of a customer who finds n π − (0|n)

I|N − customers in the system is n+1 , where πI|N (0|n) is the probability that an arriving µ + θ customer finds the server at state 0 (inactive), given that there are n customers. Therefore, the expected benefit of such a customer, if he decides to enter, is equal to

− C(n + 1) CπI|N (0|n) R− − . µ θ

(25)

− We seek equilibrium strategies of threshold type. Hence, we must compute πI|N (0|n), when all customers follow the same threshold strategy. Assume that all customers use the same threshold for entrance ne . The stationary distribution of the corresponding Markov chain is − from Proposition 2 with ne (0) = ne (1) = ne . Thus, the embedded probabilities πI|N (0|n) are equal to λp(n, 0) − (0|n) = , n = 0, 1, . . . , ne + 1, πI|N λp(n, 0) + λp(n, 1)1{n ≥ 1} where 1{n ≥ 1} is the indicator function of the set {1, 2, . . .}. Using the various forms of p(n, i) from (19)-(23) we obtain · ³ ρ ´n ´¸−1 λ+θ ³ − πI|N (0|n) = 1 + 1− , n = 0, 1, . . . ne , (26) µ−λ−θ σ · ³ ρ ´ne ´ θ ¸−1 θ λ+θ ³ − 1− + πI|N (0|ne + 1) = 1 + µµ−λ−θ σ µ · µ ¶¸−1 ³ ´ θ ρ ne +1 = 1+ 1− . (27) µ−λ−θ σ

7

In light of (25)-(27) we introduce the function · ³ ρ ´n ´¸−1 C(n + 1) C λx + θ ³ f (x, n) = R − − 1+ 1− , µ θ µ−λ−θ σ x ∈ [0, 1], n = 0, 1, 2, . . .

(28)

which will allow us to prove the existence of equilibrium threshold strategies and derive the corresponding thresholds. Let fU (n) = f (1, n), fL (n) = f (0, n), n = 0, 1, 2, . . . . It is easy to see that fU (0) = fL (0) = R − Cµ − Cθ > 0 (because of (1)). limn→∞ fU (n) = limn→∞ fL (n) = −∞. Hence, there exists nU such that

(29) In addition,

fU (0), fU (1), . . . , fU (nU ) > 0 and fU (nU + 1) ≤ 0.

(30)

The function f (x, n) is clearly increasing with respect to x for every fixed n, thus, fL (n) ≤ fU (n), n = 0, 1, 2, . . . . In particular, fL (nU + 1) ≤ 0, while fL (0) > 0. Hence there exists nL ≤ nU such that fL (nL ) > 0 and fL (nL + 1), . . . , fL (nU ), fL (nU + 1) ≤ 0. (31) We can now establish the existence of equilibrium threshold policies in the almost observable case. We have the following. Theorem 2 In the almost observable M/M/1 queue with setup times all pure threshold strategies ‘observe N (t), enter if N (t) ≤ ne and balk otherwise’, for ne = nL , nL + 1, . . . , nU , are equilibrium strategies. Proof. Consider a tagged customer at his arrival instant and assume that all other customers follow the same threshold strategy ‘observe N (t), enter if N (t) ≤ ne and balk otherwise’ for − some fixed ne ∈ {nL , nL + 1, . . . , nU }. Then, πI|N (0|n) is given by (26)-(27). If the tagged customer finds n ≤ ne customers and decides to enter, his expected net benefit is equal to · ³ ρ ´n ´¸−1 C(n + 1) C λ+θ ³ R− − 1+ 1− = fU (n) > 0, µ θ µ−λ−θ σ because of (25),(26),(28),(29) and (30). So in this case the customer prefers to enter. If the tagged customer finds n = ne + 1 customers and decides to enter, his expected net benefit is · µ ³ ρ ´ne +1 ¶¸−1 C(ne + 2) C θ R− − 1+ 1− = fL (ne + 1) ≤ 0, µ θ µ−λ−θ σ because of (25),(27),(28),(29) and (31). So in this case the customer prefers to balk. Remark 4. When nL < nU , there exist multiple equilibrium threshold strategies. This is observed generally in ‘Follow-The-Crowd’ situations, where in equilibrium customers tend to adopt the behavior of other customers. In models where the space of strategies can be ordered (e.g., when strategies can be parametrized by a single number) we have a ‘Follow-The-Crowd’ (FTC) (resp., ‘Avoid-The-Crowd’ (ATC)) situation when one’s optimal response to a strategy x adopted by all others is increasing (resp., decreasing) in x (see e.g. Hassin and Haviv (1997), and Hassin and Haviv (2003), p. 6). In the framework of our model, the first part R − C(n+1) µ 8

(1, 1) o

w O ww w µ θ ww w{ w λ / (0, 0) (1, 0)

λ µ

λqe

/

(2, 1) · · · (ne , 1) o O

O

θ λ

θ

µ

/

(ne + 1, 1) O

θ

/ (2, 0) · · · (ne , 0) λqe / (ne + 1, 0)

Figure 3: Transition rate diagram for the (ne , qe ) mixed threshold strategy of the expression (25), for the expected benefit of a tagged customer who decides to enter when he observes n customers, depends only on the number of customers that are present in front of Cπ −

(0|n)

him and not on the policy of the other customers. On the other hand, the part I/Nθ which depends on the policy adopted by the other customers is easily seen to decrease in n. We can then see, using (26)-(27), that a threshold best response of the tagged customer (whenever it exists) is increasing to the threshold policy followed by the other customers. This means that a tagged customer, whose expected net benefit is given by (25), is more willing to enter the system if the other customers use a higher threshold policy, i.e. he adopts the behavior of the others. This shows that we have an FTC situation. We can also seek equilibrium strategies in the more general class of mixed threshold strategies. A strategy of the form ‘observe N (t) and enter if N (t) ≤ ne − 1, enter with probability qe if N (t) = ne and balk otherwise’ is referred to as a mixed threshold strategy of type (ne , qe ) (also as (ne − 1 + qe )-threshold strategy, see Hassin and Haviv (2003), p. 8). To find equilibrium − strategies in this class, we must compute πI|N (0|n), when all customers follow the same mixed threshold strategy. Proposition 2 is not anymore applicable, because the transition rates of the system have changed. More specifically the transition rates all remain the same except for transitions (ne , 0) → (ne + 1, 0) and (ne , 1) → (ne + 1, 1). For these transitions the rates are now λqe instead of λ. The system is now depicted in Figure 3. The stationary analysis of this system is analogous to that in Proposition 2 for the pure threshold policy, with minor modifications. The following formulas can be derived for the − embedded probabilities πI|N (0|n). ³ ρ ´n ´¸−1 λ+θ ³ 1− , n = 0, 1, . . . ne − 1, µ−λ−θ σ · ³ ρ ´ne ´¸−1 λqe + θ ³ − πI|N (0|ne ) = 1 + 1− , µ−λ−θ σ · ³ ρ ´ne ´ θ ¸−1 θ λqe + θ ³ − πI|N (0|ne + 1) = 1 + 1− + . µµ−λ−θ σ µ − πI|N (0|n)

· = 1+

(32) (33) (34)

For the case nL < nU , the following theorem shows that there is one mixed threshold equilibrium policy between every two consecutive pure threshold equilibrium policies. Theorem 3 In the almost observable M/M/1 queue with setup times all mixed threshold strategies of the form ‘observe N (t) and enter if N (t) ≤ ne − 1, enter with probability qe if N (t) = ne and balk otherwise’ are equilibrium strategies for ne = nL + 1, . . . , nU and     1 µ−λ−θ  C ¡ ρ ¢ne qe =  − 1 − θ  . (35) C(ne +1) λ 1− σ θ(R − ) µ

9

Proof. Fix an ne ∈ {nL + 1, . . . , nU } and define the corresponding qe by (35), i.e. qe is the unique solution of f (x, ne ) = 0. The quantity qe is a probability because f (x, ne ) is continuous with respect to x and f (0, ne )f (1, ne ) = fL (ne )fU (ne ) ≤ 0, because of (30)-(31). We now consider a tagged customer at his arrival instant and assume that all other customers follow the same mixed threshold strategy (ne , qe ). If the tagged customer finds n ≤ ne − 1 customers and decides to enter, his expected net benefit is · ³ ρ ´n ´¸−1 C(n + 1) C λ+θ ³ R− − 1+ 1− = fU (n) > 0, µ θ µ−λ−θ σ because of (25), (32) and (30). Therefore, the tagged customer prefers to enter. If the tagged customer finds n = ne customers and decides to enter, his expected net benefit is equal to · ³ ρ ´ne ´¸−1 C(ne + 1) C λqe + θ ³ R− − 1+ 1− = f (qe , ne ) = 0, µ θ µ−λ−θ σ because of (25), (33) and (35). So in this case the tagged customer is indiferent between entering and balking. Any decision is optimal and in particular entering with probability qe is optimal. If the tagged customer finds n = ne + 1 customers and decides to enter, his expected net benefit is · ³ ρ ´ne ´ θ ¸−1 C(ne + 2) C θ λqe + θ ³ R− − 1+ 1− + µ θ µµ−λ−θ σ µ · ¸ ³ ³ ´ ´ θ λ+θ C(ne + 2) C ρ ne θ −1 − 1+ ≤ R− 1− + µ θ µµ−λ−θ σ µ · µ ¶¸−1 ³ ´ θ ρ ne +1 C(ne + 2) C − 1+ 1− = R− µ θ µ−λ−θ σ = fL (ne + 1) ≤ 0. Indeed, the initial expression is valid because of (25) and (34). The inequality is due to qe ≤ 1. The first equality follows from (27) and the second from with (31), since ne +1 ∈ {nL +2, . . . , nU + 1}. Thus, the customer prefers to balk. The above argument shows that any mixed threshold strategy (ne , qe ) with ne and qe satisfying the conditions of the proposition is a best response against itself, therefore, it is an equilibrium strategy. Remark 5. For an equilibrium strategy y there may be a best response z 6= y such that z is strictly a better response against itself than y is. In this case, given that the customers start with strategy y, they may all adopt strategy z and then they will never return to y. In this sense, y is ‘unstable’ or ‘transient’. If no such z exists then y is said to be an evolutionarily stable strategy (ESS). In particular, an equilibrium strategy which is a unique best response to itself is ESS. Equilibrium ESS rule out ‘unstable’ equilibrium strategies and are considered a useful refinement of the equilibrium concept. In the present model, if, as is typically the case, the inequalities in (30)-(31) are strict, then the equilibrium pure threshold strategies are ESS, since they are unique best responses against themselves. On the contrary, the equilibrium mixed threshold strategies are not ESS. Indeed, it is easy to see that if all customers start from an equilibrium threshold strategy (ne , qe ) with qe ∈ (0, 1), as in Theorem 3, and deviate to any threshold strategy (ne , q), then (ne , qe ) is not a better response for an individual against (ne , q) 10

λq(1)

(1, 1) o O

/

λq(1)

(2, 1) o O

/

λq(1)

··· o

/

λq(1)

(n, 1) o O

/

λq(1)

(n + 1, 1) o O

/

···

µ µ µ µ µ w ww θ θ θ θ ww µ w w{ wλq(0) / (1, 0) λq(0) / (2, 0) λq(0) / · · · λq(0)/ (n, 0) λq(0) / (n + 1, 0) λq(0) / · · · (0, 0)

Figure 4: Transition rate diagram for the (q(0), q(1)) mixed strategy than (ne , q) is. In the light of this observation, we can restrict our attention to pure threshold strategies only. Remark 6. A moment of reflection in the proofs of Theorems 2 and 3 shows that the only equilibrium strategies of pure/mixed threshold type are those specified there. However, there also exist equilibrium strategies of non-threshold type. For example consider the strategy ‘observe N (t) and enter if N (t) 6= ne + 1’ for some ne ∈ {nL , nL + 1, . . . , nU }. This is also an equilibrium. Indeed this strategy agrees with the ne −threshold strategy for states in {(n, i) | 0 ≤ i ≤ 1, i ≤ n ≤ ne + 1 }. On the other hand, the states in {(n, i)| 0 ≤ i ≤ 1, n > ne + 1} are transient under both strategies and therefore do not influence the arriving customer’s net benefit. These irrational equilibrium strategies can be eliminated by considering the refined concept of subgame perfect equilibrium (SPE) (for details see Hassin and Haviv (2002), and Hassin and Haviv (2003), p. 5).

4. Equilibrium mixed strategies for the unobservable cases In this section we turn our attention to the unobservable cases, where arrivals do not observe the number of customers present. We will prove that there exist equilibrium mixed strategies. We begin with the almost unobservable case in which arriving customers observe the state i of the server at their arrival instant. A mixed strategy for a customer is specified by a vector (q(0), q(1)), where q(i) is the probability of joining when the server is in state i. If all customers follow the same mixed strategy (q(0), q(1)), then the system follows a Markov chain similar to that described in Figure 1, except that the arrival rate equals λ(i) = λq(i) for states where the server is in state i. The state space Sau for the almost unobservable case is identical to the original state space S and the transition diagram is illustrated in Figure 4. Let (pau (n, i) : (n, i) ∈ S) be the stationary distribution of the corresponding system. The balance equations are presented below. p(0, 0)λ(0) = p(1, 1)µ

(36)

p(n, 0)(λ(0) + θ) = p(n − 1, 0)λ(0), n = 1, 2, . . .

(37)

p(1, 1)(λ(1) + µ) = p(1, 0)θ + p(2, 1)µ

(38)

p(n, 1)(λ(1) + µ) = p(n − 1, 1)λ(1) + p(n, 0)θ + p(n + 1, 1)µ, n = 2, 3, . . .

(39)

To obtain the stationary probabilities p(n, i) we proceed as in Proposition 2. We set ρ(0) =

λ(0) λ(1) λ(0) , ρ(1) = , σ(0) = . µ µ λ(0) + θ

By iterating (37) we obtain p(n, 0) = σ(0)n p(0, 0), n = 0, 1, . . . . 11

From (39) it follows that p(n, 1), n = 1, 2, . . . is a solution of the nonhomogeneous linear difference equation with constant coefficients µxn+1 − (λ(1) + µ)xn + λ(1)xn−1 = −θp(n, 0) = −θσ(0)n p(0, 0), n = 2, 3, . . . . Employing the standard approach (as in the proof of Proposition 1), we can find the general solution of this equation. Note that the roots of the characteristic equation are now 1 and ρ(1). For brevity we consider again only the typical case where σ(0) 6= ρ(1). Note also that σ(0) 6= 1 by definition. Using (36) and the normalization equation we obtain the appropriate constants and we have the following. Proposition 2 Consider an M/M/1 queue with setup times and σ(0) 6= ρ(1), in which customers observe the state i of the server upon arrival and enter with probability q(i), i.e., they follow the mixed policy (q(0), q(1)). The system is stable if and only if ρ(1) < 1. In this case, the stationary probabilities (pau (n, i) : (n, i) ∈ S) are pau (n, 0) = pau (n, 1) =

(1 − σ(0))(1 − ρ(1)) σ(0)n , n = 0, 1, . . . 1 − ρ(1) + ρ(0) (1 − σ(0))(1 − ρ(1))ρ(0) (σ(0)n − ρ(1)n ), n = 1, 2, . . . . (1 − ρ(1) + ρ(0))(σ(0) − ρ(1))

(40) (41)

Remark 7. Proposition 2 can be alternatively proved by employing matrix analytic methods. Indeed, by partitioning the state space as S = ∪∞ n=0 l(n), where l(0) = {(0, 0)} and l(n) = {(n, 0), (n, 1)}, n = 1, 2, . . . , we obtain a continuous time quasi-birth-death (QBD) process. If we partition its stationary distribution according to the levels by setting p¯au (0) = pau (0, 0) and p¯au (n) = (pau (n, 0), pau (n, 1)), n ≥ 1, we have that the stationary distribution is matrix geometric (see e.g. Neuts (1981) Theorem 1.7.1 in p.32). More specifically for the Markov chain of interest we have that (1 − σ(0))(1 − ρ(1)) p¯au (0) = 1 − ρ(1) + ρ(0) µ ¶ (1 − σ(0))(1 − ρ(1))σ(0) (1 − σ(0))(1 − ρ(1))ρ(0) p¯au (1) = , 1 − ρ(1) + ρ(0) 1 − ρ(1) + ρ(0) p¯au (n) = p¯au (1)Rn−1 where

à R=

σ(0) 0

θρ(0) µ(1−ρ(0))

!

ρ(1)

is the so-called rate matrix of the chain. The rate matrix R in this case is explicitly computable, because the block matrices of this QBD are upper triangular. By diagonalizing R, we derive after some algebra (40) and (41). The social benefit per time unit when all customers follow a mixed policy (qe (0), qe (1)) can now be easily computed as µ ¶ C(λ(0) + θ) C 1 − ρ(1) q(0) R − − Sau = λ 1 − ρ(1) + ρ(0) µθ θ µ ¶ ρ(0) C(λ(0) + θ) C +λ q(1) R − − . (42) 1 − ρ(1) + ρ(0) µθ µ − λ(1) 12

We now consider a customer who finds the server at state i upon arrival. His mean sojourn − − time is E[N µ|i]+1 + 1−i θ , where E[N |i] is the expected number of customers in system found by an arrival, given that the server is found at state i. The expected benefit of such a customer who decides to enter is C(E[N − |i] + 1) C(1 − i) R− − . (43) µ θ We thus need to compute E[N − |i] when all customers follow the same mixed strategy, say (q(0), q(1)). Assume that the system is stable under this strategy, i.e., λq(1) < µ. Then the − probability πN |I (n|i), that an arrival finds n customers in system, given that the server is found at state i is equal to p(n, i)λ(i) − P∞ πN , n = i, i + 1, . . . . |I (n|i) = k=i p(k, i)λ(i) P − We substitute (40)-(41) in (44) and from E[N − |i] = ∞ n=i nπN |I (n, i) we obtain E[N − |0] = E[N − |1] =

(44)

σ(0) 1 − σ(0) ρ(1) 1 + 1 − ρ(1) 1 − σ(0)

(45) (46)

Substituting into (43) we can identify mixed equilibrium strategies for the almost unobservable model. Theorem 4 In the almost unobservable M/M/1 queue with setup times and λ < µ, there exists a unique mixed equilibrium strategy (qe (0), qe (1)) ‘observe I(t) and enter with probability qe (I(t))’ where the vector (qe (0), qe (1)) is given as follows. Case I: 1θ < µ1 .  ³ ³ ´ ´ 1 µθR  − µ − θ ,0 ,  λ C       (1, 0), µ µ ¶¶ (qe (0), qe (1)) = 1 C  1, µ − ,  C(λ+θ) λ  R− µθ      (1, 1), Case II:

1 µ

≤

1 θ

≤

³

R∈ R∈ R∈ R∈

C + Cθ , C(λ+θ) + Cθ µθ hµ C(λ+θ) + Cθ , C(λ+θ) µθ µθ

h

h

C(λ+θ) µθ

+

C C(λ+θ) µ, µθ

C(λ+θ) µθ

+

C µ−λ , ∞

´

´ C µ

+

C µ−λ

1 µ−λ

´

.

1 µ−λ .

 ³ ³ ´ ´ ³ ´ µ−θ 1 µθR C C C(λ+θ) C  − µ − θ , , R ∈ + , +  λ C λ µ θ µθ θ   µ ¶¶  µ h C(λ+θ) 1 C C C(λ+θ) 1, λ µ − , R∈ + θ , µθ + (qe (0), qe (1)) = C(λ+θ) µθ R− µθ   h ´   C  (1, 1), R ∈ C(λ+θ) + , ∞ . µθ µ−λ Case III:

´

+

C µ−λ

< 1θ .

 ³ ³ ´ ´ ³ ´  1 µθR − µ − θ , 1 , R ∈ C + C , C(λ+θ) + C λ C θ µθ ´ θ hµ (qe (0), qe (1)) = C(λ+θ) C  (1, 1), R∈ + θ ,∞ . µθ 13

´

Proof. Consider a tagged customer who finds the server at state 0 upon arrival. If he decides to enter, his expected net benefit is R−

C(E[N − |0] + 1) C − µ θ

C C − µ(1 − σ(0)) θ C(λ(0) + θ) C = R− − . µθ θ = R−

Therefore we have two cases: Case 1: Cµ + Cθ < R ≤ C(λ+θ) + Cθ . In this case if all customers who find the system empty enter µθ with probability qe (0) = 1, then the tagged customer suffers a negative expected benefit if he decides to enter. Hence, qe (0) = 1 does not lead to an equilibrium. Similarly, if all customers use qe (0) = 0 then the tagged customer receives a positive benefit from entering, thus qe (0) = 0 also cannot be part of an equilibrium mixed strategy. Therefore, there exists a unique qe (0), satisfying C(λqe (0) + θ) C R− − =0 µθ θ for which customers are indifferent between entering and balking. This is given by µ ¶ 1 µθR qe (0) = −µ−θ . λ C

(47)

Case 2: C(λ+θ) + Cθ < R. In this case, for every strategy of the other customers, the tagged µθ customer has a positive expected net benefit if he decides to enter. Hence qe (0) = 1.

(48)

We next consider qe (1) and tag a customer who finds the server at state 1 upon arrival. If he decides to enter his expected net benefit is equal to R−

C(E[N − |1] + 1) µ

C C − µ(1 − ρ(1)) µ(1 − σ(0)) C C(λ(0) + θ) = R− − µ − λ(1) µθ ( C C in case 1 θ − µ−λ(1) , = C(λ+θ) C R − µ−λ(1) − µθ , in case 2. = R−

(49)

Therefore, to find qe (1) in equilibrium, we must examine Cases 1 and 2 separately and consider the following subcases in each: + Cθ and Cθ < Cµ . Case 1a: Cµ + Cθ < R ≤ C(λ+θ) µθ ¶ ¶ µ µ 1 µθR − µ − θ ,0 . (qe (0), qe (1)) = λ C Case 1b:

C µ

+

C θ

0 is geometric with parameter λqeµ(1) , therefore E[N − |N − > 0] = µ−λqµe (1) . Hence the expected net benefit from entering of an arriving customer C who finds the server active is equal to R − Cµ − µ−λq . If R < 2C µ then this quantity is always e (1) negative, thus the best response is 0 and we obtain qe (1) = 0, the first branch of (50). For C C C C 2C µ ≤ R ≤ µ + µ−λ and R > µ + µ−λ we obtain the other two branches of (50) with analogous reasoning.

Remark 8. In Theorem 4 it was assumed that λ < µ. For the opposite case an analogous result holds with the various cases simplified. Specifically, it can be shown following a similar analysis, that, when λ ≥ µ, there exists a unique mixed equilibrium strategy (qe (0), qe (1)) ‘observe I(t) and enter with probability qe (I(t))’ where the vector (qe (0), qe (1)) is given as follows. Case I: 1θ < µ1 .  ³ ³ ´ ´ ³ ´ 1 µθR C C C(λ+θ) C  − µ − θ , 0 , R ∈ + , +  λ C θ µθ θ   hµ  C(λ+θ) C C(λ+θ) + θ , µθ + (1, 0), R∈ (qe (0), qe (1)) = µθ µ ¶¶ µ  h ´   C C  , R ∈ C(λ+θ) + , ∞ .  1, λ1 µ − C(λ+θ) µθ µ R−

Case II:

1 µ

≤

C µ

´

µθ

1 θ

 ³ ³ ´ ´ ³ ´ C(λ+θ) µθR µ−θ   λ1 C − µ − θ , λ , R ∈ Cµ + Cθ , µθ + Cθ µ µ ¶¶ h ´ (qe (0), qe (1)) = 1 C C  , R ∈ C(λ+θ) + , ∞ .  1, λ µ − C(λ+θ) µθ θ R−

µθ

We finally consider the fully unobservable case, where the customers do not observe the state of the system at all. Here a mixed strategy for a customer is specified by the probability q of entering. The stationary distribution of the system state is given by Proposition 2 by taking q(0) = q(1) = q. The equilibrium behavior of the customers is described in the following. Theorem 5 In the fully unobservable M/M/1 queue with setup times and λ < µ, there exists a unique mixed equilibrium strategy ‘enter with probability qe ’, where qe is given by  µ ¶ ³ ´   1 µ − CC , R ∈ C + C , C + C λ µ θ µ−λ θ R− θ (51) qe = h ´  C  1, R ∈ µ−λ + Cθ , ∞ .

16

Proof. We consider a tagged customer at his arrival instant. If he decides to enter his expected net benefit is C C R − (E[N − ] + 1) − Pr[I − = 0], (52) µ θ where N − represents the number of customers in the system and I − the state of the server seen by ρ σ the tagged customer. Using Proposition 2 we obtain E[N − ] = 1−ρ + 1−σ and Pr[I − = 0] = 1−ρ, with ρ =

λq µ,

σ=

λq λq+µ .

Hence, the expected net benefit in (52) is equal to R−

³ When R ∈

C µ

+

C C θ , µ−λ

+

C θ

C C − . µ − λq θ

(53)

´

, we find that (53) has a unique root in (0, 1) which gives the first ´ C branch of (54). When R ∈ µ−λ + Cθ , ∞ the quantity in (53) is positive for every q, thus the best response is 1 and the unique equilibrium point is qe = 1, which gives the second branch of (54). h

Remark 9. For the case λ ≥ µ the analog of Theorem 5 is that there exists a unique mixed equilibrium strategy ‘enter with probability qe ’, where à ! 1 C qe = µ− , (54) λ R − Cθ ³ for R ∈

C µ

+

´ .

C θ ,∞

5. Numerical results In this section we present numerical experiments that show the effect of the information level as well as several parameters on the behavior of the system. Specifically, we are interested in the values of the equilibrium thresholds for the observable models and the values of the equilibrium entrance probabilities for the unobservable models as well as the social benefit per unit time when the customers follow equilibrium strategies. We first consider the fully and almost observable systems and explore the sensitivity of the equilibrium pure threshold policies with respect to the service reward R, arrival rate λ and setup rate θ (by rescaling if necessary, we can assume without loss of generality that µ = C = 1). The results are presented in Figure 5. An interesting conjecture arises from this figure. We observe that in all three diagrams the range of thresholds for the almost observable case, {nL , . . . , nU } is contained inside the range between ne (0) and ne (1) for the fully observable case. In other words, in the almost observable model the common threshold has an intermediate value between the two separate thresholds that the customers use when they are given the additional information on the server state Regarding the sensitivity in specific parameters, we can make the following observations. When the service reward R varies (above the minimum level Cµ + Cθ required for entrance), the thresholds increase in a linear fashion, up to the integrality requirement. Under varying arrival rate λ, the fully observable thresholds remain fixed. This is expected from Theorem 1, since the arrival rate is irrelevant to the customer’s decision when he has full state information. On the other hand, the almost observable threshold range increases with λ. This means that when an arriving customer is given information on the number of customers present, then he is more likely 17

to enter the system when the arrival rate is higher. The reason is that when the arrival rate is high, it is more likely that the server is active, therefore the expected delay from server activation is reduced, while the delay due to the customers already present is not affected. Finally, when the setup rate θ varies, all thresholds increase, except for ne (1) which remains constant. This is certainly intuitive, because when the server activation is faster customers generally have a greater incentive to enter both in the fully and the almost observable case. In the second numerical experiment we turn to the almost and fully unobservable systems and explore the sensitivity of the equilibrium entrance probabilities. The results are shown in Figure 6. A general observation from this Figure is that, similarly to the previous experiment,the entrance probability in the fully unobservable model is always inside the interval formed by the two entrance probabilities in the almost unobservable case. However the relative ordering of qe (0) and qe (1) varies. Therefore, when customers are not given the information about server state, they follow an intermediate strategy and join the queue with a probability between those that they would employ for the two separate cases, were these known. With regard to sensitivity, the entrance probabilities are increasing with respect to R, which is intuitive. Furthermore, they are nonincreasing with respect to λ, therefore customers are less inclined to enter the system as the arrival rate increases. This is in contrast to the observable cases, where thresholds are nondecreasing in λ. The reason for the difference is that here the information on the number in the system is not available, therefore, when λ is higher, arriving customers expect that the system is more loaded and are less willing to enter. With respect to the setup rate θ, the behavior of the entrance probabilities varies. While for the most part they are all increasing with θ, which is intuitive, there is a range of small values of θ in which qe (1) is decreasing. The last numerical experiment is concerned with the social benefit under the equilibrium strategy for the different information levels. The results are presented in Figures 7, 8, 9. In the almost observable case we have seen that in general there are multiple pure equilibrium strategies, corresponding to thresholds nL , nL + 1, . . . , nU . For this reason we present the social benefit under the two extreme threshold values, nL , nU . In the figures we observe that the difference in social benefit is small between the fully and almost observable case, while there may be significant differences between the observable and the unobservable models. Thus, it may be argued that the customers as a whole are generally better off when upon arrival they are given the information on how many are already present and are left to decide whether to join the system or not, while the additional information on the server state is not very beneficial. In regard to the sensitivity in parameters, the social benefit is increasing with respect to the reward R and setup rate θ, both of which are intuitive. Regarding the arrival rate λ, the social benefit achieves a maximum for intermediate values of this parameter. The reason for this behavior is that when the arrival rate is small the system is rarely crowded, therefore as more customers arrive they are served and the social benefit improves. However as λ continues to increase a smaller percentage decides to enter and those who do are subjected to longer delays, which has a detrimental effect on the social benefit. 6. Conclusions and Extensions In this paper we considered the problem of analyzing customer behavior in equilibrium, in an M/M/1 queue with server setups where customers decide whether to join the system upon arrival. We identified four cases with respect to the level of information provided to arriving customers and derived the equilibrium strategies for each case. We also discussed the sensitivity with respect to various parameters as well as the effect of the information level on the social benefit. The focus of this work was on equilibrium analysis. The social benefit for each information 18

level was determined, under the assumption that all customers decide independently and rationally whether to enter the system. On the other hand, one can think of situations where a central planner employs acceptance policies that maximize the social benefit, under the various levels of information on the system state. As is expected, such plans, which do not generally constitute equilibrium policies, may not be possible to enforce when customers are allowed to make independent decisions. However they are applicable when all customers are potentially willing to enter the queue without necessarily considering benefit versus costs, perhaps because of other hard constraints. Such situations may arise among others in traffic engineering and telecommunications. Another interesting extension would be to incorporate the present model in a profit maximizing framework, where the owner or manager of the system imposes an entrance fee. The problem is to find the optimal fee that maximizes the owner’s total profit subject to the constraint that customers independently decide whether to enter or not and take into accout the fee, in addition to the service reward and waiting cost. On a higher layer of optimization, one might also consider problems of the optimal capacity level that maximizes profit, given that the entrance pricing is designed optimally. Acknowledgements The authors acknowledge support by EPEAEKII-Pythagoras, grant # 70/3/7388 (European social fund and national resources). A. Burnetas was also supported by grant ELKE/70/4/7584 (University of Athens). A. Economou was also supported by grants ELKE/70/4/6415 (University of Athens) and MTM2005-01248 (Spanish research project).

References [1] Adan, I. and J. van der Wal (1998) “Difference and Differential Equations in Stochastic Operations Research”, Online Notes, URL: http://www.win.tue.nl/ iadan/ [2] Artalejo, J.R. and M.J. Lopez-Herrero (2003) On the M/M/m queue with removable servers, in: Stochastic Point Processes, eds. S.K. Srinivasan and A. Vijayakumar (Narosa Publishing House) 124-143. [3] Artalejo, J.R., A. Economou and M.J. Lopez-Herrero (2005) Analysis of a multiserver queue with setup times, Queueing Systems 52, 53-76. [4] Bischof, W. (2001) Analysis of M/G/1-queues with setup times and vacations under six different service disciplines, Queueing Systems 39, 265-301. [5] Borthakur, A. and G. Choudhury (1999) A multiserver Poisson queue with a general startup time under N -Policy, Calcutta Statistical Association Bulletin 49, 199-213. [6] Chen, H. and M. Frank (2004) Monopoly pricing when customers queue, IIE Transactions 36, 569-581. [7] Chen, H. and M. Frank (2001) State dependent pricing with a queue, IIE Transactions 33, 847-860. [8] Choudhury, G. (1998) On a batch arrival poisson queue with a random setup and vacation period, Computers and Operations Research 25, 1013-1026. [9] Choudhury, G. (2000) An MX /G/1 queueing system with a setup period and a vacation period, Queueing Systems 36, 23-38. [10] Edelson, N. M. and K. Hildebrand (1975) Congestion tolls for Poisson queueing processes, Econometrica 43, 81-92. [11] Elaydi, S. N. (1999) An Introduction to Difference Equations, Mathematics. Springer-Verlag, New York.

19

[12] Hassin, R. (1986) Consumer information in markets with random products quality: The case of queues and balking, Econometrica 54, 1185-1195. [13] Hassin, R. and M. Haviv (1997) Equilibrium threshold strategies: the case of queues with priorities, Operations Research 45, 966-973. [14] Hassin, R. and M. Haviv (2002) Nash equilibrium and subgame perfection in observable queues, Annals of Operations Research 113, 15-26. [15] Hassin, R. and M. Haviv (2003) To Queue or Not to Queue: Equilibrium Behavior in Queueing Systems, Kluwer Academic Publishers, Boston. [16] He, Q.M. and E. Jewkes (1995) Flow time in the MAP/G/1 queue with customer batching and setup times, Stochastic Models 11, 691-711. [17] Naor, P. (1969) The regulation of queue size by levying tolls, Econometrica 37,15-24. [18] Takagi, H. (1991) Queueing Analysis - A Foundation of Performance Evaluation, Vol. 1: Vacation and Priority Systems, New York, North-Holland. [19] Tian, N. and Z.G.Zhang (2006) Vacation Queueing Models: Theory and Applications, SpringerVerlag, New York. [20] Yechiali, U. (1971) On optimal balking rules and toll charges in the GI/M/1 queue, Operations Research 19, 349-370.

20

(a) 40

n (0) e ne(1) n L n

thresholds

30

U

20 10 0 20

25

30 R

35

40

(b) 25

ne(0) ne(1) nL nU

thresholds

20 15 10 5 0

0

0.5

1

λ

1.5

(c) 25

n (0) e n (1) e nL n

thresholds

20 15

U

10 5 0

0

0.1

0.2 θ

0.3

0.4

Figure 5: Equilibrium Thresholds for Observable and Almost Observable Systems. Sensitivity with Respect to: (a) R, for λ = 0.4, µ = 1, θ = 0.05, C = 1 ; (b) λ, for µ = 1, θ = 0.05, C = 1, R = 25; (c) θ, for λ = 0.8µ = 1, C = 1, R = 25

21

(a)

Entrance Prob.

1

q(0) q(1) qe

0.8 0.6 0.4 0.2 0

5

10

15

20

R (b)

Entrance Prob.

1

q(0) q(1) qe

0.8 0.6 0.4 0.2 0

0

0.2

0.4

0.6

λ

0.8

1

(c)

Entrance Prob.

1

q(0) q(1) qe

0.8 0.6 0.4 0.2 0

0

0.1

0.2 θ

0.3

0.4

Figure 6: Equilibrium Entrance Probabilities for Unobservable and Almost Unobservable Systems. Sensitivity with Respect to: (a) R, for λ = 0.9, µ = 1, θ = 0.15, C = 1 ; (b) λ, for µ = 1, θ = 0.15, C = 1, R = 8; (c) θ, for λ = 0.9, µ = 1, C = 1, R = 10

22

F.O. A.O. nL A.O. nU

15

A.U. F.U

S0

10

5

0 20

25

30

35 R

40

45

50

Figure 7: Social Benefit for Different Information Levels. Sensitivity with Respect to: (a) R, for λ = 0.5, µ = 1, θ = 0.05, C = 1

23

F.O. A.O. nL A.O. nU A.U. F.U

10

S0

8

6

4

2

0

0

0.2

0.4

0.6

0.8

1 λ

1.2

1.4

1.6

1.8

2

Figure 8: Social Benefit for Different Information Levels. Sensitivity with Respect to λ, for µ = 1, θ = 0.05, C = 1, R = 25

24

F.O. A.O. nL A.O. nU

8

A.U. F.U

7

6

S0

5

4

3

2

1

0

0

0.1

0.2

0.3

0.4

0.5 θ

0.6

0.7

0.8

0.9

1

Figure 9: Social Benefit for Different Information Levels. Sensitivity with Respect to θ, for λ = 0.9, µ = 1, C = 1, R = 10

25