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Equilibrium analysis of the observable queues with balking and delayed repairs
Applied Mathematics and Computation 218 (2011) 2716–2729
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Equilibrium analysis of the observable queues with balking and delayed repairs Jinting Wang ⇑, Feng Zhang Department of Mathematics, Beijing Jiaotong University, Beijing, China
a r t i c l e
i n f o
Keywords: Game theory M/M/1 queue Balking strategy Server breakdown Delayed repair
a b s t r a c t The equilibrium threshold balking strategies are investigated for the fully observable and partially observable single-server queues with server breakdowns and delayed repairs. Upon arriving, the customers decide whether to join or balk the queue based on observation of the queue length and status of the server, along with the consideration of waiting cost and the reward after finishing their service. By using Markov chain approach and system cost analysis, we obtain the stationary distribution of queue size of the queueing systems and provide algorithms in order to identify the equilibrium strategies for the fully and partially observable models. Finally, the equilibrium threshold balking strategies and the equilibrium social benefit for all customers are derived for the fully and partially observable system respectively, both with server breakdowns and delayed repairs. Ó 2011 Elsevier Inc. All rights reserved.
1. Introduction Recently, Economou and Kanta [2] studied the equilibrium balking strategies in the observable M/M/1 queue with an unreliable server and repairs. They considered the system with two different levels of information: (1) fully observable case. The customers can observe both the queue length and the state of the server upon arrival; (2) almost observable case. The customers can observe only the queue length and the state of the server is unobservable. Based on the information available, the customers decide whether to join or balk the system. It is assumed that as soon as the server fails it instantaneously undergoes repairs. Actually, such systems that require repairs after server’s breakdowns are very common in practice, especially in manufacturing industries, communication networks, and numerous practical problems arising from the day-to-day life situations, see [1,12], among others. However, in many real-life situations it may not be feasible to start the repair process immediately due to non-availability of the repair facility and therefore the system may delay the repair time. In this paper, we investigate such a queueing system based on Economou and Kanta [2] but assume that the failed server will experience an exponential delayed time before it begins repair process. Under this assumption, we study the equilibrium threshold balking strategies for both of the fully and almost observable single-server queues. The literature on the effect of information on the strategic behavior in queueing systems begins with Naor [10], who studied an M/M/1 queueing model with a linear reword-cost structure in the fully observable case. It is assumed that an arriving customer observes the number of customers and then makes his decision whether to join or balk. Edelson and Hildebrand [3] investigated Naor’s model by assuming that there is no information on the queue length for an arriving customer. They mentioned that it is possible for a revenue-maximizing service provider to make wrong decision about service capacity, either in the service rate or the number of servers. However, these effects disappear when balking is not allowed. ⇑ Corresponding author. E-mail address: [email protected] (J. Wang). 0096-3003/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2011.08.012
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Besides, several other papers have dealt with the value of information in congested systems, some of which have proved that information may reduce customers’ welfare. Hassin [5] showed that it may be, but is not always, socially optimal to prevent suppression of information on actual queue length, and that it is never optimal to encourage suppression when the profit maximizing server prefers to reveal the length of the queue. That is, the server may sometimes prefer less information to more. Guo and Zipkin [4] considered a queue with balking under three levels of delayed information: no information, system occupancy, and exact waiting time. They showed how to compute the key performance measures in the three systems and compare them. Hassin [6] analyzed the effect of information and uncertainty on profits in an unobservable single server queueing system. He obtained explicit answers to the effect of information about the system’s parameters on the server profits and system’s overall welfare. Hassin and Haviv [7] dealed with the economic analysis of a queueing model with priorities in which two priority levels can be purchased and obtain all of the Nash equilibrium strategies (pure or mixed) of the threshold type. General conditions for optimality of threshold type policies for controlling Poisson input–output systems were given by Hassin and Henig [9]. The fundamental results about various observable models can be found in the comprehensive monographs of Hassin and Haviv [8] and Stidham [11] with extensive bibliographical references. To the best of the authors’ knowledge, studies for the equilibrium behavior of customers in queueing system with server’s delayed repair process do not yet exist. It is the aim of this paper to study the equilibrium behavior of customers in the context of an observable M/M/1 queue with an unreliable server and delayed repairs. The paper is organized as follows. Descriptions of the model and price structure are given in Section 2. In Section 3, the equilibrium strategies for fully observable and partially observable queues are identified and the equilibrium social benefit for both models are derived. In Section 4, some numerical examples are presented to illustrate the effect of several parameters on the customers’ behavior in the considered models. Finally, in Section 5, some conclusions are given. 2. Model description We consider the fully and partially observable single-server queue with an infinite waiting room where customers arrive according to a Poisson process with rate k and service times are assumed to be exponentially distributed with rate l. The server has an exponential lifetime with failure rate f when he is working. Once the server fails it will experience an exponential delayed time to begin the repair process. During the delay time, the server stops providing service to arriving customers and waits for repair facility to begin the repair. The repair times are assumed to be exponentially distributed with parameter h. That is, due to non-availability of the repair facility, the repair process may not be started immediately when the server fails. We introduce the repair delayed time as the time interval between the epoch of server breakdown and the beginning of repair process. It is assumed that the delayed time is an independent and identically distributed random variable D with an exponential distribution function D(t) = 1 eat, t P 0. Thus, we can represent the state of the system at time t by a pair (N(t), I(t)), where N(t) and I(t) denote the number of customers and the state of the server (r: under repair; s: working state; d: delay period) respectively. The process {(N(t), I(t)), t P 0} is a two-dimensional continuous time Markov chain with non-zero transition rates given by
qðn;iÞðnþ1;iÞ ¼ k; qðn;sÞðn1;sÞ ¼ l;
n ¼ 0; 1; 2; . . . ; i ¼ r; s; d; n ¼ 1; 2; 3; . . . ;
qðn;rÞðn;sÞ ¼ h;
n ¼ 0; 1; 2; . . . ;
qðn;sÞðn;dÞ ¼ f;
n ¼ 0; 1; 2; . . . ;
qðn;dÞðn;rÞ ¼ a;
n ¼ 0; 1; 2; . . . :
As in Economou and Kanta [2], we assume that the customers are allowed to decide whether to join or balk upon their arrival based on the information they have. After service, every customer receives a reward of R units. This may reflect his satisfaction or the added value of being served. On the other hand, there exists a waiting cost of C units per time unit when the customers remains in the system including the time of waiting in queue and being served. Customers are risk neutral and maximize their expected net benefit. Their decisions are irrevocable that retrials of balking customers and reneging of entering customers are not allowed. Each customer can observe the number of customers ahead of him upon his arrival. In this paper, we consider two information cases regarding whether the customers observe also the state of the server or not, which are called fully observable and partially observable cases in the literature. 3. Equilibrium threshold strategies In this section we shall find that there exist equilibrium strategies of threshold type in two cases mentioned above. In the fully observable case where customers are informed both the state of the server I(t) and the number of the present customer N(t) at arrival time t, a pure threshold strategy is specified by a triple (ne(r), ne(s), ne(d)) and the balking strategy has the form ‘While arriving at time t, observe (N(t), I(t)); enter if N(t) 6 ne(I(t)) 1 and balk otherwise’. In the partially observable case, a pure threshold strategy is specified by a single number ne and has the form ‘While arriving at time t, observe N(t); enter if N(t) 6 ne 1 and balk otherwise’. These two cases will be studied separately regarding how much the system information are known to customers just arriving.
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Fig. 1. Transition rate diagram for the (ne(r), ne(s), ne(d)) equilibrium threshold strategy in the fully observable queue with breakdowns and delayed repairs.
3.1. Fully observable queue Fig. 1 illustrates the dynamics of the queueing system in the fully observable case, where customers are informed both the state of the server I(t) and the number of the present customer N(t) at arrival time t. And we have the following result. Theorem 1. In the fully observable M/M/1 queue with breakdowns and delayed repairs there exist a triple of thresholds:
ðne ðrÞ; ne ðsÞ; ne ðdÞÞ ¼
ðRh CÞla Rhla ðRha Ch CaÞl ; ; ; C ðah þ hf þ afÞ Cðah þ hf þ afÞ Cðah þ hf þ afÞ
ð1Þ
such that a customer who observes the system at state (N(t), I(t)) upon his arrival enters if N(t) 6 ne(I(t)) 1 and balks otherwise. Proof. It is obvious that for an arriving customer, his expected net reward if he enters is:
Sðn; iÞ ¼ R CTðn; iÞ;
ð2Þ
where T(n, i) denotes his expected mean sojourn time given that he finds the system at state ((N(t), I(t)) upon his arrival. Then we have the following equations:
1 þ Tðn; sÞ; n ¼ 0; 1; 2; . . . ; h l 1 f Tð0; sÞ ¼ þ Tð0; dÞ; lþf l lþf 1 l f þ Tðn 1; sÞ þ Tðn; dÞ; Tðn; sÞ ¼ lþf lþf lþf 1 Tðn; dÞ ¼ þ Tðn; rÞ; n ¼ 0; 1; 2; . . . :
Tðn; rÞ ¼
ð3Þ ð4Þ n ¼ 1; 2; 3; . . . ;
a
ð5Þ ð6Þ
Solving the system of (3) and (6) for n = 0 along with (4) we obtain T(0, r), T(0, s) and T(0, d). From Eqs. (3) and (6) we can obtain:
Tðn; dÞ ¼
1
a
þ
1 þ Tðn; sÞ; h
n ¼ 0; 1; 2; . . . :
ð7Þ
By plugging (7) in (5) and using T(0, 1) we get:
f f 1 Tðn; sÞ ¼ ðn þ 1Þ 1 þ þ ; a h l
n ¼ 0; 1; 2; . . . :
ð8Þ
By plugging (8) in (3) we have:
f f 1 1 Tðn; rÞ ¼ ðn þ 1Þ 1 þ þ þ ; a h l h
n ¼ 0; 1; 2; . . . :
ð9Þ
Solving Eq. (6) by using (9), we obtain:
f f 1 1 1 Tðn; dÞ ¼ ðn þ 1Þ 1 þ þ þ þ ; a h l h a
n ¼ 0; 1; 2; . . . :
ð10Þ
If S(n, i) > 0, i.e. if the reward for service exceeds the expected cost for waiting, the customer prefers to enter. And if S(n, i) = 0, i.e., if the reward is equal to the cost he is indifferent between entering and balking. By solving S(n, i) P 0 for n, using Eqs. (2)
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and (8), (9), (10), we obtain that the customer arriving at time t decides to enter if and only if n 6 ne(I(t)) 1 where (ne (r), ne(s), ne(d)) is given by (1). This strategy is preferable, independently of what the other customers do, i.e. it is a weakly dominant strategy. h For the stationary analysis of the system, note that if all customers follow the threshold strategy in (1) the system follows a Markov chain with state space restricted to Qfo = {(n, i)j0 6 n 6 ne(s), i = r, s, d} and identical transition rates. The corresponding stationary distribution {p(n, i): (n, i) 2 Qfo} is obtained as the unique positive normalized solution of the following system of balance equations:
ðk þ hÞpð0; rÞ ¼ apð0; dÞ;
ð11Þ
ðk þ hÞpðn; rÞ ¼ kpðn 1; rÞ þ apðn; dÞ;
n ¼ 1; 2; . . . ; ne ðrÞ 1;
ð12Þ
hpðne ðrÞ; rÞ ¼ kpðne ðrÞ 1; rÞ þ apðne ðrÞ; dÞ;
ð13Þ
hpðn; rÞ ¼ apðn; dÞ;
ð14Þ
n ¼ ne ðrÞ þ 1; . . . ; ne ðsÞ;
ðk þ fÞpð0; sÞ ¼ lpð1; sÞ þ hpð0; rÞ;
ð15Þ
ðk þ f þ lÞpðn; sÞ ¼ lpðn þ 1; sÞ þ hpðn; rÞ þ kpðn 1; sÞ;
n ¼ 1; 2; . . . ; ne ðsÞ 1;
ðf þ lÞpðne ðsÞ; sÞ ¼ hpðne ðsÞ; rÞ þ kpðne ðsÞ 1; sÞ;
ð16Þ ð17Þ
ðk þ aÞpð0; dÞ ¼ fpð0; sÞ;
ð18Þ
ðk þ aÞpðn; dÞ ¼ kpðn 1; dÞ þ fpðn; sÞ;
n ¼ 1; 2; . . . ; ne ðdÞ 1;
ð19Þ
apðne ðdÞ; dÞ ¼ fpðne ðdÞ; sÞ þ kpðne ðdÞ 1; dÞ; apðn; dÞ ¼ fpðn; sÞ; n ¼ ne ðdÞ þ 1; . . . ; ne ðsÞ:
ð20Þ ð21Þ
Denote by
p1 ¼ pðne ðsÞ; sÞ;
ð22Þ
p2 ¼ pðne ðrÞ; rÞ;
ð23Þ
p3 ¼ pðne ðdÞ; dÞ;
ð24Þ
and define:
k
q¼ : l From Eqs. (14), (16), (17) and (21), we have:
pðn; sÞ ¼ qðne ðsÞnÞ p1 ; n ¼ ne ðrÞ; . . . ; ne ðsÞ; f pðn; rÞ ¼ qðne ðsÞnÞ p1 ; n ¼ ne ðrÞ þ 1; . . . ; ne ðsÞ; h f pðn; dÞ ¼ qðne ðsÞnÞ p1 ; n ¼ ne ðrÞ þ 1; . . . ; ne ðsÞ:
ð25Þ ð26Þ ð27Þ
a
Considering Eqs. (12), (13), (16) and (21):
lpðn; sÞ ¼ kpðn 1; sÞ þ kpðn 1; rÞ; n ¼ ne ðdÞ þ 1; . . . ; ne ðrÞ:
ð28Þ
From Eqs. (12), (13), (21) and (28), it is easy to obtain that {p(n, r): n = ne(d), . . . , ne(r)} [ {p(n, s): n = ne(d), . . . , ne(r) 1} [ {p(n, d): n = ne(d) + 1, . . . , ne(r)} can be expressed by p1 and p2. Again, with the help of Eqs. (12), (16), (19) and (20), we observe that p(n, r),p(n, s) and p(n, d) are functions of p1, p2 and p3 when n = 0,1, . . . , ne(d) 1. Plugging p(0, r), p(0, s), p(0, d), and p(1, s) in (11), (15) and (18), p2 and p3 can be expressed as the functions of p1. Thus we can express all stationary probabilities in terms of p1. The remaining probability, p1, can be found from the normalization equation: ne ðsÞ X
pðn; rÞ þ
n¼0
n e ðsÞ X
pðn; sÞ þ
n¼0
n e ðsÞ X
pðn; dÞ ¼ 1:
n¼0
Pne ðsÞ Pne ðsÞ Because the probability of balking is n¼n pðn; rÞ þ pðne ðsÞ; sÞ þ n¼n pðn; dÞ, the social benefit per time unit when all e ðrÞ e ðdÞ customers follow the threshold policy (ne(r),ne(s),ne(d)) given in Theorem 1 equals:
0 SBfo ¼ kR@1
n e ðsÞ X n¼ne ðrÞ
pðn; rÞ pðne ðsÞ; sÞ
ne ðsÞ X n¼ne ðdÞ
1 pðn; dÞA C
n e ðsÞ X n¼0
nðpðn; rÞ þ pðn; sÞ þ pðn; dÞÞ:
ð29Þ
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3.2. Partially observable queue In this section we proceed to the partially observable case where the arriving customers observe the number of customers upon arrival, but not the state of the server. The transition diagram is depicted in Fig. 2. The corresponding equilibrium strategies within the class of pure threshold strategies will be studied. To this end, it is necessary to obtain the stationary distribution of the system when the customers follow a given pure threshold strategy. We have the following preliminary result. Lemma 1. In the partially observable M/M/1 queue with breakdowns and delayed repairs where the customers enter the system according to a threshold strategy ‘while arriving at time t, observe N(t); enter if N(t) 6 ne 1 and balk otherwise’, the stationary distribution (P(n, i): (n, i) 2 {0,1, 2, . . . , ne} {r, s, d}) is given as follows:
Pðn; rÞ ¼ w1 qn1 þ w2 qn2 þ w3 qn3 þ w4 qn4 ; Pðn; sÞ ¼ v 1 qn1 þ v 2 qn2 þ v 3 qn3 þ v 4 qn4 ;
n ¼ 0; 1; . . . ; ne 1; n ¼ 0; 1; . . . ; ne 1;
Pðn; dÞ ¼ c1 qn1 þ c2 qn2 þ c3 qn3 þ c4 qn4 ; n ¼ 0; 1; . . . ; ne 1; " # 4 4 X X k ne 1 ne 1 ðci þ v i þ wi Þqi l v i qi ; Pðne ; rÞ ¼ ðf þ lÞ lh i¼1 i¼1 Pðne ; sÞ ¼ Pðne ; dÞ ¼
4 k X
l
ðci þ v i þ wi Þqni e 1 ;
4 k f X
a l
ðci þ v i þ wi Þqni e 1 þ
i¼1
ð32Þ ð33Þ ð34Þ
i¼1
"
ð30Þ ð31Þ
4 X
# ci qine 1 ;
ð35Þ
i¼1
where
vi ¼
ðk þ aÞqi k ci ; fqi
i ¼ 1; 2; 3; 4;
vi
k ðk þ f þ l lqi Þ; i ¼ 1; 2; 3; 4; h qi ðk þ hÞ½ðk þ f þ lÞðk þ aÞ þ kl klðk þ aÞ afh a¼ ; lðk þ hÞðk þ aÞ lðk þ hÞðk þ aÞ kðk þ hÞð2k þ a þ f þ lÞ k½ðk þ f þ lÞðk þ aÞ þ kl þ ; b¼ lðk þ hÞðk þ aÞ lðk þ hÞðk þ aÞ wi ¼
ð36Þ ð37Þ ð38Þ ð39Þ
2
c¼ d¼
k ð3k þ a þ f þ h þ lÞ ; lðk þ hÞðk þ aÞ
k3 : lðk þ hÞðk þ aÞ
ð40Þ ð41Þ
and ci(i = 1,2,3,4) are determined by using normalization condition and qi(i = 1, 2, 3, 4) are four roots of equation x4 + ax3 + bx2 + cx + d = 0. Proof. The stationary distribution (P(n, i)) is obtained by using the balance equations:
Fig. 2. Transition rate diagram for the ne equilibrium threshold strategy in the partially observable queue with breakdowns and delayed repairs.
J. Wang, F. Zhang / Applied Mathematics and Computation 218 (2011) 2716–2729
ðk þ hÞPð0; rÞ ¼ aPð0; dÞ;
2721
ð42Þ
ðk þ hÞPðn; rÞ ¼ kPðn 1; rÞ þ aPðn; dÞ;
n ¼ 1; 2; . . . ; ne 1;
ð43Þ
hPðne ; rÞ ¼ kPðne 1; rÞ þ aPðne ; dÞ;
ð44Þ
ðk þ fÞPð0; sÞ ¼ lPð1; sÞ þ hPð0; rÞ;
ð45Þ
ðk þ f þ lÞPðn; sÞ ¼ lPðn þ 1; sÞ þ hPðn; rÞ þ kPðn 1; sÞ;
n ¼ 1; 2; . . . ; ne 1;
ðf þ lÞPðne ; sÞ ¼ hPðne ; rÞ þ kPðne 1; sÞ;
ð46Þ ð47Þ
ðk þ aÞPð0; dÞ ¼ fPð0; sÞ;
ð48Þ
ðk þ aÞPðn; dÞ ¼ kPðn 1; dÞ þ fPðn; sÞ;
n ¼ 1; 2; . . . ; ne 1;
ð49Þ
aPðne ; dÞ ¼ fPðne ; sÞ þ kPðne 1; dÞ:
ð50Þ
From Eqs. (43), (46) and (49) we obtain:
lðk þ hÞðk þ aÞPðn þ 1; dÞ þ fðk þ hÞ½ðk þ f þ lÞðk þ aÞ þ kl þ klðk þ aÞ afhgPðn; dÞ k½ðk þ hÞð2k þ a þ f þ lÞ þ ðk þ f þ lÞðk þ aÞ þ klPðn 1; dÞ þ k2 ð3k þ a þ f þ h þ lÞPðn 2; dÞ k3 Pðn 3; dÞ ¼ 0;
n ¼ 3; 4; . . . ; ne 2:
ð51Þ
which is a four-order difference equation with solution qi (i = 1, 2, 3, 4). Therefore, we can set:
Pðn; dÞ ¼ c1 qn1 þ c2 qn2 þ c3 qn3 þ c4 qn4 ;
n ¼ 0; 1; . . . ; ne 1;
ð52Þ
where ci, i = 1, 2, 3, 4 are constants to be determined. By plugging (52) in (49), we get:
Pðn; sÞ ¼ v 1 qn1 þ v 2 qn2 þ v 3 qn3 þ v 4 qn4 ;
n ¼ 1; 2; . . . ; ne 1:
ð53Þ
Again, by plugging (53) in (46), we get:
Pðn; rÞ ¼ w1 qn1 þ w2 qn2 þ w3 qn3 þ w4 qn4 ;
n ¼ 2; 3; . . . ; ne 2
ð54Þ
By putting n = 1 in (46) and using (53) and (54), it is followed that:
Pð0; sÞ ¼ v 1 þ v 2 þ v 3 þ v 4 : Then we obtain (31) by combining (53) and the above equation. In a similar manner, putting n = 1, n = 2 and n = ne 1 in (43), and from the above equations, we get (30) with the help of (51) and (54). The results (33)–(35) can be derived from (44), (47) and (50) by using Eqs. (30)–(32). By plugging (30)–(32) in (42), (45) and (48), we have:
( 4 X ðk þ hÞ½ðk þ aÞqi kðk þ f þ l lqi qk Þ i
hfqi
i¼1
) a ci ¼ 0;
ð55Þ
4 X ðlqi kÞ½ðk þ aÞqi k ci ¼ 0; 2 fqi i¼1 4 X ci i¼1
qi
ð56Þ
¼ 0:
ð57Þ
with the help of Eqs. (55)–(57), then the value of ci, i = 1, 2, 3, 4 can be determined by using normalization condition. h We now proceed to find the expected net reward of a customer that observes n customers ahead of him and decides to enter. We have the following: Lemma 2. Consider the partially observable M/M/1 queue with breakdowns and delayed repairs where the customers enter to the system according to a threshold strategy ‘While arriving at time t, observe N(t); enter if N(t) 6 ne 1 and balk otherwise’. The net benefit of a customer that observes n customers and decides to enter is given by
! P4 P4 f f 1 1h i¼1 wi qni þ ð1h þ a1Þ i¼1 ci qni ; SðnÞ ¼ R C ðn þ 1Þ 1 þ þ þ P4 n a h l i¼1 ðci þ v i þ wi Þqi 1 Pðne ; rÞ þ 1h þ a1 Pðne ; dÞ f f 1 Sðne Þ ¼ R C ðne þ 1Þ 1 þ þ þ h : a h l Pðne ; rÞ þ Pðne ; sÞ þ Pðne ; dÞ
n ¼ 0; 1; . . . ; ne 1;
ð58Þ ð59Þ
Proof. The expected net reward, if he enters, for a customer that observes n customers is:
SðnÞ ¼ R CTðnÞ;
ð60Þ
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where T(n) = E[SjN = n] denotes his expected mean sojourn time given that he finds n customers in the system just before his arrival. We let Pr (I = ijN = n) be the probability that the state of the server is i when he observes n customers in the system upon his arrival. Conditioning on the state of the server that he finds upon arrival and taking into account (3) and (6) we obtain:
1 1 1 PrðI ¼ djN ¼ nÞ PrðI ¼ rjN ¼ nÞ þ Tðn; sÞ þ þ TðnÞ ¼ Tðn; sÞPrðI ¼ sjN ¼ nÞ þ Tðn; sÞ þ h h a 1 1 1 PrðI ¼ djN ¼ nÞ; ¼ Tðn; sÞ þ PrðI ¼ rjN ¼ nÞ þ þ h h a
ð61Þ
where T(n, 1) is given by (8) and
kPðn; rÞ ; kPðn; rÞ þ kPðn; sÞ þ kPðn; dÞ kPðn; dÞ PrðI ¼ djN ¼ nÞ ¼ ; kPðn; rÞ þ kPðn; sÞ þ kPðn; dÞ
PrðI ¼ rjN ¼ nÞ ¼
n ¼ 0; 1; . . . ; ne :
Using the stationary probabilities obtained in Lemma 1 we obtain the probabilities Pr (I = rjN = n) and Pr (I = djN = n) for n = 0,1, . . . , ne. So, we have:
P4 P4 f f 1 1h i¼1 wi qni þ 1h þ a1 i¼1 ci qni TðnÞ ¼ ðn þ 1Þ 1 þ þ þ ; P4 n a h l i¼1 ðc i þ v i þ wi Þqi 1 Pðne ; rÞ þ 1h þ a1 Pðne ; dÞ f f 1 Tðne Þ ¼ ðne þ 1Þ 1 þ þ þ h ; a h l Pðne ; rÞ þ Pðne ; sÞ þ Pðne ; dÞ
n ¼ 0; 1; . . . ; ne 1;
ð62Þ ð63Þ
where P(ne, r),P(ne, s) and P(ne, d) are given by Eqs. (33)–(35). And then, Eqs. (58) and (59) can be obtained by plugging Eqs. (62) and (63) in (60) for every ne. It should be noted that a customer does not enter the system even if he finds no customers in front of him if S(0) < 0, otherwise, he enters the queue. h Next, we describe the equilibrium balking pure threshold strategies in the partially observable case by assuming S(0) > 0. We have the following result. Theorem 2. Define the sequences (f1(n): n = 0, 1, . . .) and (f2 (n):n = 0, 1, . . .) by
(
) P4 P4 f f 1 1h i¼1 wi qni þ 1h þ a1 i¼1 ci qni ; f1 ðnÞ ¼ R C ðn þ 1Þ 1 þ þ þ P4 n a h l i¼1 ðci þ v i þ wi Þqi
n ¼ 0; 1; . . . ;
ð64Þ
(
f f 1 1 f2 ðnÞ ¼ R C ðn þ 1Þ 1 þ þ þ P4 a h l Wn 1h P4i¼1 v i qn1 þ 1a i¼1 ci qn1 i i !) " # 4 4 4 X fþl f 1 1 X 1 1 1 1X n1 n1 n1 þ ð c þ v þ w Þ q v q þ c q ; þ þ i i i i i i i i h a a i¼1 lh2 la h a i¼1 h2 i¼1
n ¼ 0; 1; . . . ;
ð65Þ
P4 l f 1 n1 where Wn ðfþ lh þ l þ laÞ i¼1 ðc i þ v i þ wi Þqi . By definition, f1(n) = S(n),n = 0,1, . . . , ne 1, f2(ne) = S(ne). Moreover, f1(n) > f2(n), n = 0,1, . . .. Then there exist finite non-negative integers nL 6 nU such that:
f1 ð0Þ; f1 ð1Þ; . . . ; f1 ðnU 1Þ > 0;
f 1 ðnU Þ 6 0;
ð66Þ
and
f2 ðnU Þ; f2 ðnU 1Þ; f2 ðnU 2Þ; . . . ; f2 ðnL Þ 6 0; f2 ðnL 1Þ > 0;
ð67Þ
f2 ðnU Þ; f2 ðnU 1Þ; f2 ðnU 2Þ; . . . ; f2 ð0Þ 6 0:
ð68Þ
or
In the partially observable M/M/1 queue with breakdowns and delayed repairs the pure threshold strategies of the form ‘While arriving at time t, observe N(t); enter if N(t) 6 ne 1 and balk otherwise’, for ne 2 {nL, nL + 1, . . . , nU}, are equilibrium strategies. Proof. We have that f1(0) > 0 if we assume that S(0) > 0 and limn!1 f1 ðnÞ ¼ 1, so if nU is the subscript of the first nonpositive term of the sequence (f1(n)), we have that for the finite number nU the condition (66) holds.
J. Wang, F. Zhang / Applied Mathematics and Computation 218 (2011) 2716–2729
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On the other hand, f1(n) > f2(n),n = 0,1, . . . . In particular we conclude that f2(nU) < f1(nU) 6 0. Now, we begin to go backwards, starting from the subscript nU, towards 0 and we let nL 1 be the subscript of the first positive term of the sequence (f2(n)). Then we have (67). If all the terms of (f2(n)) going backwards from nU to 0 are non-positive we have (68). We can now establish the existence of equilibrium threshold policies in the partially observable case. In this model we consider a tagged customer at his arrival instant and assume that all other customers follow the same threshold strategy ‘while arriving at time t, observe N(t), enter if N(t) 6 ne 1 and balk otherwise’ for some fixed ne 2 {nL, nL + 1, . . . , nU}. If the tagged customer finds n 6 ne 1 customers in front of him and decides to enter, his expected benefit is equal to f1(n) > 0 because of (60), (64) and (66). So in this case the customer prefers to enter. If the tagged customer finds n = ne customers in front of him and decides to enter, his expected benefit is equal to f2(ne) 6 0 because of (60), (65), (67) or (68). So in this case the customer prefers to balk. h Define
PðnÞ ¼ Pðn; rÞ þ Pðn; sÞ þ Pðn; dÞ;
n ¼ 1; 2; . . . ; ne :
Because the probability of balking is equal to P(ne), the social benefit per time unit when all customers follow the threshold policies ne given in Lemma 2 equals:
SBpo ¼ kRð1 Pðne ÞÞ C
ne X
nPðnÞ:
ð69Þ
n¼0
Remark 1. Theorem 2 provides an algorithm in order to identify the equilibrium strategies of the partially observable model. One has to start computing (f1(n)) up to the first negative term. This ‘forward’ procedure yields the highest equilibrium threshold nU. Then, one has to start computing (f2(n)) starting from f2(nU) and going towards 0 till the first positive term. This ‘backward’ procedure yields the lowest equilibrium threshold nL. Remark 2. When nL < nU, there exist multiple equilibrium threshold strategies such that ‘While arriving at time t, observe N(t), enter if N(t) 6 ne 2; enter with probability qe if N(t) = ne 1 and balk otherwise’ between every two consecutive equilibrium pure threshold strategies. We have a ‘Follow-The-Crowd’ (FTC) situation when one’s optimal response to a strategy x adopted by all others is increasing in x. In our model, let we consider the function Sne ðnÞ given by (60), for the expected benefit of a tagged customer who decides to enter when he observes n customers in front of him while the others follow the threshold strategy ne. Now suppose that the other customers adopt the strategy ne + 1. Then the tagged customer’s benefit function Sne þ1 ðnÞ coincides with Sne ðnÞ for all n = 0,1, . . . , ne 1 while Sne þ1 ðne Þ ¼ f1 ðne Þ > f2 ðne Þ ¼ Sne ðne Þ. We can then see that a threshold best response of the tagged customer when the others follow the strategy ne + 1 is greater than his best response threshold when the others follow the strategy ne. This means that a tagged customer, whose expected net benefit is given by (60), 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. In a conclusion, the outcome we get in the partially observable case agrees with the results in [7] and [8].
4. Numerical experiments In this section, we show the effect of several parameters on the behavior of customers both in the fully observable and partially observable queueing system with server breakdowns and delayed repairs by some numerical examples. We explore the sensitivity of the equilibrium thresholds and the equilibrium social benefits with respect to some main indicators of the system. In Fig. 3 we observe that thresholds are monotonically decreasing functions of breakdown rate f. This is because frequent breakdowns of the server increase the waiting time of customer and their payment, which makes them reluctant to enter. Figs. 4 and 5 show that if a customer sees the high delayed repair rate a or high repair rate h upon his arrival he prefers to join the queue due to the fact that the period of delayed repair or repair will not last much time. Fig. 6 depicts the equilibrium thresholds versus service rate l. It is intuitive that an arriving customer is more likely to enter when the server can serve more customers per time. Compare Fig. 7 with Fig. 8 we can get some conclusions. The higher reward R the customers gain from service, the higher the waiting cost they can afford. So the thresholds increase with R. However, if waiting costs customers more money, they may be not willing to wait for service avoiding paying too much. Moreover, we can observe that in all first six figures equilibrium thresholds {nL, nL + 1, . . . , nU} for the partially observable case always locate between ne(d) and ne(s) for the fully observable case. In other words, the thresholds in the partially model have intermediate values between the two separate thresholds provided that the state of the server can be observed by arriving customers. Figs. 9–12 are concerned with the social benefit under the equilibrium strategy for the different information levels. For the partially observable case we present the social benefit under the two extreme thresholds nL and nU. In regard to the sensitivity of social benefit in equilibrium, we observe that it increases with respect to a, h and R, whereas decreases with respect to the breakdown rate f, which are intuitive. Furthermore, the four figures show that the social benefit of all customers
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J. Wang, F. Zhang / Applied Mathematics and Computation 218 (2011) 2716–2729 22 n (r) e
20
n (s) e
n (d) e
18
n
L
n
U
16
Thresholds
14
12
10
8
6
4
2 0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0.45
0.5
ζ Fig. 3. Thresholds vs. f for k = 0.5, l = 1, a = 0.3, h = 0.2, C = 1, R = 30.
18 n (r) e
16
n (s) e
n (d) e
n
14
L
n
U
Thresholds
12
10
8
6
4
2
0 0.05
0.1
0.15
0.2
0.25
α
0.3
0.35
0.4
Fig. 4. Thresholds vs. a for k = 0.5, l = 1, f = 0.1, h = 0.2, C = 1, R = 30.
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20 n (r) e
18
n (s) e
n (d) e
16
n
L
n
U
14
Thresholds
12
10
8
6
4
2
0 0.05
0.1
0.15
0.2
0.25
0.3
θ
0.35
0.4
0.45
0.5
Fig. 5. Thresholds vs. h for k = 0.5, l = 1, a = 0.3, f = 0.1, C = 1, R = 30.
90 n (r) e
80
n (s) e
n (d) e
70
n
L
n
U
Thresholds
60
50
40
30
20
10
0 0.5
1
1.5
2
2.5
3
3.5
4
μ Fig. 6. Thresholds vs. l for k = 0.5, a = 0.3, f = 0.1, h = 0.2, C = 1, R = 30.
4.5
5
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20 n (r) e
n (s) e
18
n (d) e
n
L
n
Thresholds
16
U
14
12
10
8
6 20
22
24
26
28
30
32
34
36
38
R Fig. 7. Thresholds vs. R for k = 0.5, l = 1, a = 0.3, f = 0.1, h = 0.2, C = 1.
16 n (r) e
n (s) e
14
n (d) e
n
L
n
Thresholds
12
U
10
8
6
4
2
1
1.1
1.2
1.3
1.4
1.5
1.6
1.7
C Fig. 8. Thresholds vs. C for k = 0.5, l = 1, a = 0.3, f = 0.1, h = 0.2, R = 30.
1.8
1.9
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11 P.O.nL
10
P.O.nU
9
F.O.
Equilibrium social benefit
8
7
6
5
4
3
2
1 0.05
0.1
0.15
0.2
0.25
ζ
0.3
0.35
0.4
0.45
0.5
Fig. 9. Equilibrium social benefit for different information levels vs. f for k = 0.5, l = 1, a = 0.3, h = 0.2, C = 1, R = 30.
9
8
P.O.n
L
P.O.n
U
7
Equilibrium social benefit
F.O.
6
5
4
3
2
1 0.05
0.1
0.15
0.2
0.25
α
0.3
0.35
0.4
0.45
0.5
Fig. 10. Equilibrium social benefit for different information levels vs. a for k = 0.5, l = 1, f = 0.1, h = 0.2, C = 1, R = 30.
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10
P.O.n
L
9
P.O.n
U
Equilibrium social benefit
8
F.O.
7
6
5
4
3
2 0.05
0.1
0.15
0.2
0.25
θ
0.3
0.35
0.4
0.45
0.5
Fig. 11. Equilibrium social benefit for different information levels vs. h for k = 0.5, l = 1, a = 0.3, f = 0.1, C = 1, R = 30.
10
P.O.n
L
9 P.O.n
U
F.O.
Equilibrium social benefit
8
7
6
5
4
3 20
22
24
26
28
30
32
34
36
38
R Fig. 12. Equilibrium social benefit for different information levels vs. R for k = 0.5, l = 1, a = 0.3, f = 0.1, h = 0.2, C = 1.
J. Wang, F. Zhang / Applied Mathematics and Computation 218 (2011) 2716–2729
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in the fully observable case is not always more than that in the partially observable case. This reveals that the information about the system is not always helpful for increasing social benefit in equilibrium. That is, in some cases more accurate information improves the equilibrium social benefit. However, in other cases information can actually hurt the customers. 5. Conclusion Inspired by Economou and Kanta [2], we studied equilibrium balking strategies of the observable single-server queue with breakdowns and delayed repairs. The equilibrium threshold balking strategies and equilibrium social benefits for all customers are derived respectively for the fully and partially observable systems with server breakdowns and delayed repairs. It is the first time that server’s delayed repair process is taken into account for the effect of system information on the strategic behavior in queueing systems. Algorithms are also provided to identify the equilibrium strategies for the fully observable system and partially observable system. We observe by numerical examples that the thresholds in the partially observable system take intermediate values between two extremes, i.e., two separate thresholds in the fully observable case. ‘‘Follow-The-Crowd’’ (FTC) property has also been observed. Moreover, it is shown that the information about the system is not always helpful for increasing social benefit in equilibrium. In some cases more accurate information improves the equilibrium social benefit, but in other cases can hurt the customers. A new property in this model is that the repair time has two stages and hence it is not memoryless. So, the analysis on the Nash equilibria of the present model with delayed repairs is more general and practical than the model with immediate repair. Acknowledgements The authors thank the anonymous referee’s comments leading to a much improved presentation. This work was supported by the National Natural Science Foundation of China (Nos. 10871020 and 11010301053). References [1] B. Avi-Itzhak, P. Naor, Some queueing problems with the service station subject to breakdown, Operations Research 10 (1962) 303–320. [2] A. Economou, S. Kanta, Equilibrium balking strategies in the observable single-server queue with breakdowns and repairs, Operations Research Letters 36 (2008) 696–699. [3] N.M. Edelson, K. Hildebrand, Congestion tolls for Poisson queueing processes, Econometrica 43 (1975) 81–92. [4] P.F. Guo, P. Zipkin, Analysis and comparison of queues with different levels of delay information, Management Science 53 (2007) 962–970. [5] R. Hassin, Consumer information in markets with random product quality: the case of queues and balking, Econometrica 54 (1986) 1185–1195. [6] R. Hassin, Information and uncertainty in a queueing system, Probablity in the Engineering and Informational Sciences 21 (2007) 361–380. [7] R. Hassin, M. Haviv, Equilibrium threshold strategies: the case of queues with priorities, Operations Research 45 (1997) 966–973. [8] R. Hassin, M. Haviv, To Queue or Not to Queue: Equilibrium Behavior in Queueing Systems, Kluwer Academic Publishers, Boston, 2003. [9] R. Hassin, M. Henig, Control of arrivals and departures in a state-dependent input–output system, Operations Research Letters 5 (1986) 33–36. [10] P. Naor, The regulation of queue size by levying tolls, Econometrica 37 (1969) 15–24. [11] S. Stidham Jr., Optimal Design of Queueing Systems, CRC Press, Taylor and Francis Group, Boca Raton, 2009. [12] J. Wang, J. Cao, Q. Li, Reliability analysis of the retrial queue with server breakdowns and repairs, Queueing Systems 38 (2001) 363–380.
Contents lists available at SciVerse ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Equilibrium analysis of the observable queues with balking and delayed repairs Jinting Wang ⇑, Feng Zhang Department of Mathematics, Beijing Jiaotong University, Beijing, China
a r t i c l e
i n f o
Keywords: Game theory M/M/1 queue Balking strategy Server breakdown Delayed repair
a b s t r a c t The equilibrium threshold balking strategies are investigated for the fully observable and partially observable single-server queues with server breakdowns and delayed repairs. Upon arriving, the customers decide whether to join or balk the queue based on observation of the queue length and status of the server, along with the consideration of waiting cost and the reward after finishing their service. By using Markov chain approach and system cost analysis, we obtain the stationary distribution of queue size of the queueing systems and provide algorithms in order to identify the equilibrium strategies for the fully and partially observable models. Finally, the equilibrium threshold balking strategies and the equilibrium social benefit for all customers are derived for the fully and partially observable system respectively, both with server breakdowns and delayed repairs. Ó 2011 Elsevier Inc. All rights reserved.
1. Introduction Recently, Economou and Kanta [2] studied the equilibrium balking strategies in the observable M/M/1 queue with an unreliable server and repairs. They considered the system with two different levels of information: (1) fully observable case. The customers can observe both the queue length and the state of the server upon arrival; (2) almost observable case. The customers can observe only the queue length and the state of the server is unobservable. Based on the information available, the customers decide whether to join or balk the system. It is assumed that as soon as the server fails it instantaneously undergoes repairs. Actually, such systems that require repairs after server’s breakdowns are very common in practice, especially in manufacturing industries, communication networks, and numerous practical problems arising from the day-to-day life situations, see [1,12], among others. However, in many real-life situations it may not be feasible to start the repair process immediately due to non-availability of the repair facility and therefore the system may delay the repair time. In this paper, we investigate such a queueing system based on Economou and Kanta [2] but assume that the failed server will experience an exponential delayed time before it begins repair process. Under this assumption, we study the equilibrium threshold balking strategies for both of the fully and almost observable single-server queues. The literature on the effect of information on the strategic behavior in queueing systems begins with Naor [10], who studied an M/M/1 queueing model with a linear reword-cost structure in the fully observable case. It is assumed that an arriving customer observes the number of customers and then makes his decision whether to join or balk. Edelson and Hildebrand [3] investigated Naor’s model by assuming that there is no information on the queue length for an arriving customer. They mentioned that it is possible for a revenue-maximizing service provider to make wrong decision about service capacity, either in the service rate or the number of servers. However, these effects disappear when balking is not allowed. ⇑ Corresponding author. E-mail address: [email protected] (J. Wang). 0096-3003/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2011.08.012
J. Wang, F. Zhang / Applied Mathematics and Computation 218 (2011) 2716–2729
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Besides, several other papers have dealt with the value of information in congested systems, some of which have proved that information may reduce customers’ welfare. Hassin [5] showed that it may be, but is not always, socially optimal to prevent suppression of information on actual queue length, and that it is never optimal to encourage suppression when the profit maximizing server prefers to reveal the length of the queue. That is, the server may sometimes prefer less information to more. Guo and Zipkin [4] considered a queue with balking under three levels of delayed information: no information, system occupancy, and exact waiting time. They showed how to compute the key performance measures in the three systems and compare them. Hassin [6] analyzed the effect of information and uncertainty on profits in an unobservable single server queueing system. He obtained explicit answers to the effect of information about the system’s parameters on the server profits and system’s overall welfare. Hassin and Haviv [7] dealed with the economic analysis of a queueing model with priorities in which two priority levels can be purchased and obtain all of the Nash equilibrium strategies (pure or mixed) of the threshold type. General conditions for optimality of threshold type policies for controlling Poisson input–output systems were given by Hassin and Henig [9]. The fundamental results about various observable models can be found in the comprehensive monographs of Hassin and Haviv [8] and Stidham [11] with extensive bibliographical references. To the best of the authors’ knowledge, studies for the equilibrium behavior of customers in queueing system with server’s delayed repair process do not yet exist. It is the aim of this paper to study the equilibrium behavior of customers in the context of an observable M/M/1 queue with an unreliable server and delayed repairs. The paper is organized as follows. Descriptions of the model and price structure are given in Section 2. In Section 3, the equilibrium strategies for fully observable and partially observable queues are identified and the equilibrium social benefit for both models are derived. In Section 4, some numerical examples are presented to illustrate the effect of several parameters on the customers’ behavior in the considered models. Finally, in Section 5, some conclusions are given. 2. Model description We consider the fully and partially observable single-server queue with an infinite waiting room where customers arrive according to a Poisson process with rate k and service times are assumed to be exponentially distributed with rate l. The server has an exponential lifetime with failure rate f when he is working. Once the server fails it will experience an exponential delayed time to begin the repair process. During the delay time, the server stops providing service to arriving customers and waits for repair facility to begin the repair. The repair times are assumed to be exponentially distributed with parameter h. That is, due to non-availability of the repair facility, the repair process may not be started immediately when the server fails. We introduce the repair delayed time as the time interval between the epoch of server breakdown and the beginning of repair process. It is assumed that the delayed time is an independent and identically distributed random variable D with an exponential distribution function D(t) = 1 eat, t P 0. Thus, we can represent the state of the system at time t by a pair (N(t), I(t)), where N(t) and I(t) denote the number of customers and the state of the server (r: under repair; s: working state; d: delay period) respectively. The process {(N(t), I(t)), t P 0} is a two-dimensional continuous time Markov chain with non-zero transition rates given by
qðn;iÞðnþ1;iÞ ¼ k; qðn;sÞðn1;sÞ ¼ l;
n ¼ 0; 1; 2; . . . ; i ¼ r; s; d; n ¼ 1; 2; 3; . . . ;
qðn;rÞðn;sÞ ¼ h;
n ¼ 0; 1; 2; . . . ;
qðn;sÞðn;dÞ ¼ f;
n ¼ 0; 1; 2; . . . ;
qðn;dÞðn;rÞ ¼ a;
n ¼ 0; 1; 2; . . . :
As in Economou and Kanta [2], we assume that the customers are allowed to decide whether to join or balk upon their arrival based on the information they have. After service, every customer receives a reward of R units. This may reflect his satisfaction or the added value of being served. On the other hand, there exists a waiting cost of C units per time unit when the customers remains in the system including the time of waiting in queue and being served. Customers are risk neutral and maximize their expected net benefit. Their decisions are irrevocable that retrials of balking customers and reneging of entering customers are not allowed. Each customer can observe the number of customers ahead of him upon his arrival. In this paper, we consider two information cases regarding whether the customers observe also the state of the server or not, which are called fully observable and partially observable cases in the literature. 3. Equilibrium threshold strategies In this section we shall find that there exist equilibrium strategies of threshold type in two cases mentioned above. In the fully observable case where customers are informed both the state of the server I(t) and the number of the present customer N(t) at arrival time t, a pure threshold strategy is specified by a triple (ne(r), ne(s), ne(d)) and the balking strategy has the form ‘While arriving at time t, observe (N(t), I(t)); enter if N(t) 6 ne(I(t)) 1 and balk otherwise’. In the partially observable case, a pure threshold strategy is specified by a single number ne and has the form ‘While arriving at time t, observe N(t); enter if N(t) 6 ne 1 and balk otherwise’. These two cases will be studied separately regarding how much the system information are known to customers just arriving.
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Fig. 1. Transition rate diagram for the (ne(r), ne(s), ne(d)) equilibrium threshold strategy in the fully observable queue with breakdowns and delayed repairs.
3.1. Fully observable queue Fig. 1 illustrates the dynamics of the queueing system in the fully observable case, where customers are informed both the state of the server I(t) and the number of the present customer N(t) at arrival time t. And we have the following result. Theorem 1. In the fully observable M/M/1 queue with breakdowns and delayed repairs there exist a triple of thresholds:
ðne ðrÞ; ne ðsÞ; ne ðdÞÞ ¼
ðRh CÞla Rhla ðRha Ch CaÞl ; ; ; C ðah þ hf þ afÞ Cðah þ hf þ afÞ Cðah þ hf þ afÞ
ð1Þ
such that a customer who observes the system at state (N(t), I(t)) upon his arrival enters if N(t) 6 ne(I(t)) 1 and balks otherwise. Proof. It is obvious that for an arriving customer, his expected net reward if he enters is:
Sðn; iÞ ¼ R CTðn; iÞ;
ð2Þ
where T(n, i) denotes his expected mean sojourn time given that he finds the system at state ((N(t), I(t)) upon his arrival. Then we have the following equations:
1 þ Tðn; sÞ; n ¼ 0; 1; 2; . . . ; h l 1 f Tð0; sÞ ¼ þ Tð0; dÞ; lþf l lþf 1 l f þ Tðn 1; sÞ þ Tðn; dÞ; Tðn; sÞ ¼ lþf lþf lþf 1 Tðn; dÞ ¼ þ Tðn; rÞ; n ¼ 0; 1; 2; . . . :
Tðn; rÞ ¼
ð3Þ ð4Þ n ¼ 1; 2; 3; . . . ;
a
ð5Þ ð6Þ
Solving the system of (3) and (6) for n = 0 along with (4) we obtain T(0, r), T(0, s) and T(0, d). From Eqs. (3) and (6) we can obtain:
Tðn; dÞ ¼
1
a
þ
1 þ Tðn; sÞ; h
n ¼ 0; 1; 2; . . . :
ð7Þ
By plugging (7) in (5) and using T(0, 1) we get:
f f 1 Tðn; sÞ ¼ ðn þ 1Þ 1 þ þ ; a h l
n ¼ 0; 1; 2; . . . :
ð8Þ
By plugging (8) in (3) we have:
f f 1 1 Tðn; rÞ ¼ ðn þ 1Þ 1 þ þ þ ; a h l h
n ¼ 0; 1; 2; . . . :
ð9Þ
Solving Eq. (6) by using (9), we obtain:
f f 1 1 1 Tðn; dÞ ¼ ðn þ 1Þ 1 þ þ þ þ ; a h l h a
n ¼ 0; 1; 2; . . . :
ð10Þ
If S(n, i) > 0, i.e. if the reward for service exceeds the expected cost for waiting, the customer prefers to enter. And if S(n, i) = 0, i.e., if the reward is equal to the cost he is indifferent between entering and balking. By solving S(n, i) P 0 for n, using Eqs. (2)
J. Wang, F. Zhang / Applied Mathematics and Computation 218 (2011) 2716–2729
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and (8), (9), (10), we obtain that the customer arriving at time t decides to enter if and only if n 6 ne(I(t)) 1 where (ne (r), ne(s), ne(d)) is given by (1). This strategy is preferable, independently of what the other customers do, i.e. it is a weakly dominant strategy. h For the stationary analysis of the system, note that if all customers follow the threshold strategy in (1) the system follows a Markov chain with state space restricted to Qfo = {(n, i)j0 6 n 6 ne(s), i = r, s, d} and identical transition rates. The corresponding stationary distribution {p(n, i): (n, i) 2 Qfo} is obtained as the unique positive normalized solution of the following system of balance equations:
ðk þ hÞpð0; rÞ ¼ apð0; dÞ;
ð11Þ
ðk þ hÞpðn; rÞ ¼ kpðn 1; rÞ þ apðn; dÞ;
n ¼ 1; 2; . . . ; ne ðrÞ 1;
ð12Þ
hpðne ðrÞ; rÞ ¼ kpðne ðrÞ 1; rÞ þ apðne ðrÞ; dÞ;
ð13Þ
hpðn; rÞ ¼ apðn; dÞ;
ð14Þ
n ¼ ne ðrÞ þ 1; . . . ; ne ðsÞ;
ðk þ fÞpð0; sÞ ¼ lpð1; sÞ þ hpð0; rÞ;
ð15Þ
ðk þ f þ lÞpðn; sÞ ¼ lpðn þ 1; sÞ þ hpðn; rÞ þ kpðn 1; sÞ;
n ¼ 1; 2; . . . ; ne ðsÞ 1;
ðf þ lÞpðne ðsÞ; sÞ ¼ hpðne ðsÞ; rÞ þ kpðne ðsÞ 1; sÞ;
ð16Þ ð17Þ
ðk þ aÞpð0; dÞ ¼ fpð0; sÞ;
ð18Þ
ðk þ aÞpðn; dÞ ¼ kpðn 1; dÞ þ fpðn; sÞ;
n ¼ 1; 2; . . . ; ne ðdÞ 1;
ð19Þ
apðne ðdÞ; dÞ ¼ fpðne ðdÞ; sÞ þ kpðne ðdÞ 1; dÞ; apðn; dÞ ¼ fpðn; sÞ; n ¼ ne ðdÞ þ 1; . . . ; ne ðsÞ:
ð20Þ ð21Þ
Denote by
p1 ¼ pðne ðsÞ; sÞ;
ð22Þ
p2 ¼ pðne ðrÞ; rÞ;
ð23Þ
p3 ¼ pðne ðdÞ; dÞ;
ð24Þ
and define:
k
q¼ : l From Eqs. (14), (16), (17) and (21), we have:
pðn; sÞ ¼ qðne ðsÞnÞ p1 ; n ¼ ne ðrÞ; . . . ; ne ðsÞ; f pðn; rÞ ¼ qðne ðsÞnÞ p1 ; n ¼ ne ðrÞ þ 1; . . . ; ne ðsÞ; h f pðn; dÞ ¼ qðne ðsÞnÞ p1 ; n ¼ ne ðrÞ þ 1; . . . ; ne ðsÞ:
ð25Þ ð26Þ ð27Þ
a
Considering Eqs. (12), (13), (16) and (21):
lpðn; sÞ ¼ kpðn 1; sÞ þ kpðn 1; rÞ; n ¼ ne ðdÞ þ 1; . . . ; ne ðrÞ:
ð28Þ
From Eqs. (12), (13), (21) and (28), it is easy to obtain that {p(n, r): n = ne(d), . . . , ne(r)} [ {p(n, s): n = ne(d), . . . , ne(r) 1} [ {p(n, d): n = ne(d) + 1, . . . , ne(r)} can be expressed by p1 and p2. Again, with the help of Eqs. (12), (16), (19) and (20), we observe that p(n, r),p(n, s) and p(n, d) are functions of p1, p2 and p3 when n = 0,1, . . . , ne(d) 1. Plugging p(0, r), p(0, s), p(0, d), and p(1, s) in (11), (15) and (18), p2 and p3 can be expressed as the functions of p1. Thus we can express all stationary probabilities in terms of p1. The remaining probability, p1, can be found from the normalization equation: ne ðsÞ X
pðn; rÞ þ
n¼0
n e ðsÞ X
pðn; sÞ þ
n¼0
n e ðsÞ X
pðn; dÞ ¼ 1:
n¼0
Pne ðsÞ Pne ðsÞ Because the probability of balking is n¼n pðn; rÞ þ pðne ðsÞ; sÞ þ n¼n pðn; dÞ, the social benefit per time unit when all e ðrÞ e ðdÞ customers follow the threshold policy (ne(r),ne(s),ne(d)) given in Theorem 1 equals:
0 SBfo ¼ kR@1
n e ðsÞ X n¼ne ðrÞ
pðn; rÞ pðne ðsÞ; sÞ
ne ðsÞ X n¼ne ðdÞ
1 pðn; dÞA C
n e ðsÞ X n¼0
nðpðn; rÞ þ pðn; sÞ þ pðn; dÞÞ:
ð29Þ
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J. Wang, F. Zhang / Applied Mathematics and Computation 218 (2011) 2716–2729
3.2. Partially observable queue In this section we proceed to the partially observable case where the arriving customers observe the number of customers upon arrival, but not the state of the server. The transition diagram is depicted in Fig. 2. The corresponding equilibrium strategies within the class of pure threshold strategies will be studied. To this end, it is necessary to obtain the stationary distribution of the system when the customers follow a given pure threshold strategy. We have the following preliminary result. Lemma 1. In the partially observable M/M/1 queue with breakdowns and delayed repairs where the customers enter the system according to a threshold strategy ‘while arriving at time t, observe N(t); enter if N(t) 6 ne 1 and balk otherwise’, the stationary distribution (P(n, i): (n, i) 2 {0,1, 2, . . . , ne} {r, s, d}) is given as follows:
Pðn; rÞ ¼ w1 qn1 þ w2 qn2 þ w3 qn3 þ w4 qn4 ; Pðn; sÞ ¼ v 1 qn1 þ v 2 qn2 þ v 3 qn3 þ v 4 qn4 ;
n ¼ 0; 1; . . . ; ne 1; n ¼ 0; 1; . . . ; ne 1;
Pðn; dÞ ¼ c1 qn1 þ c2 qn2 þ c3 qn3 þ c4 qn4 ; n ¼ 0; 1; . . . ; ne 1; " # 4 4 X X k ne 1 ne 1 ðci þ v i þ wi Þqi l v i qi ; Pðne ; rÞ ¼ ðf þ lÞ lh i¼1 i¼1 Pðne ; sÞ ¼ Pðne ; dÞ ¼
4 k X
l
ðci þ v i þ wi Þqni e 1 ;
4 k f X
a l
ðci þ v i þ wi Þqni e 1 þ
i¼1
ð32Þ ð33Þ ð34Þ
i¼1
"
ð30Þ ð31Þ
4 X
# ci qine 1 ;
ð35Þ
i¼1
where
vi ¼
ðk þ aÞqi k ci ; fqi
i ¼ 1; 2; 3; 4;
vi
k ðk þ f þ l lqi Þ; i ¼ 1; 2; 3; 4; h qi ðk þ hÞ½ðk þ f þ lÞðk þ aÞ þ kl klðk þ aÞ afh a¼ ; lðk þ hÞðk þ aÞ lðk þ hÞðk þ aÞ kðk þ hÞð2k þ a þ f þ lÞ k½ðk þ f þ lÞðk þ aÞ þ kl þ ; b¼ lðk þ hÞðk þ aÞ lðk þ hÞðk þ aÞ wi ¼
ð36Þ ð37Þ ð38Þ ð39Þ
2
c¼ d¼
k ð3k þ a þ f þ h þ lÞ ; lðk þ hÞðk þ aÞ
k3 : lðk þ hÞðk þ aÞ
ð40Þ ð41Þ
and ci(i = 1,2,3,4) are determined by using normalization condition and qi(i = 1, 2, 3, 4) are four roots of equation x4 + ax3 + bx2 + cx + d = 0. Proof. The stationary distribution (P(n, i)) is obtained by using the balance equations:
Fig. 2. Transition rate diagram for the ne equilibrium threshold strategy in the partially observable queue with breakdowns and delayed repairs.
J. Wang, F. Zhang / Applied Mathematics and Computation 218 (2011) 2716–2729
ðk þ hÞPð0; rÞ ¼ aPð0; dÞ;
2721
ð42Þ
ðk þ hÞPðn; rÞ ¼ kPðn 1; rÞ þ aPðn; dÞ;
n ¼ 1; 2; . . . ; ne 1;
ð43Þ
hPðne ; rÞ ¼ kPðne 1; rÞ þ aPðne ; dÞ;
ð44Þ
ðk þ fÞPð0; sÞ ¼ lPð1; sÞ þ hPð0; rÞ;
ð45Þ
ðk þ f þ lÞPðn; sÞ ¼ lPðn þ 1; sÞ þ hPðn; rÞ þ kPðn 1; sÞ;
n ¼ 1; 2; . . . ; ne 1;
ðf þ lÞPðne ; sÞ ¼ hPðne ; rÞ þ kPðne 1; sÞ;
ð46Þ ð47Þ
ðk þ aÞPð0; dÞ ¼ fPð0; sÞ;
ð48Þ
ðk þ aÞPðn; dÞ ¼ kPðn 1; dÞ þ fPðn; sÞ;
n ¼ 1; 2; . . . ; ne 1;
ð49Þ
aPðne ; dÞ ¼ fPðne ; sÞ þ kPðne 1; dÞ:
ð50Þ
From Eqs. (43), (46) and (49) we obtain:
lðk þ hÞðk þ aÞPðn þ 1; dÞ þ fðk þ hÞ½ðk þ f þ lÞðk þ aÞ þ kl þ klðk þ aÞ afhgPðn; dÞ k½ðk þ hÞð2k þ a þ f þ lÞ þ ðk þ f þ lÞðk þ aÞ þ klPðn 1; dÞ þ k2 ð3k þ a þ f þ h þ lÞPðn 2; dÞ k3 Pðn 3; dÞ ¼ 0;
n ¼ 3; 4; . . . ; ne 2:
ð51Þ
which is a four-order difference equation with solution qi (i = 1, 2, 3, 4). Therefore, we can set:
Pðn; dÞ ¼ c1 qn1 þ c2 qn2 þ c3 qn3 þ c4 qn4 ;
n ¼ 0; 1; . . . ; ne 1;
ð52Þ
where ci, i = 1, 2, 3, 4 are constants to be determined. By plugging (52) in (49), we get:
Pðn; sÞ ¼ v 1 qn1 þ v 2 qn2 þ v 3 qn3 þ v 4 qn4 ;
n ¼ 1; 2; . . . ; ne 1:
ð53Þ
Again, by plugging (53) in (46), we get:
Pðn; rÞ ¼ w1 qn1 þ w2 qn2 þ w3 qn3 þ w4 qn4 ;
n ¼ 2; 3; . . . ; ne 2
ð54Þ
By putting n = 1 in (46) and using (53) and (54), it is followed that:
Pð0; sÞ ¼ v 1 þ v 2 þ v 3 þ v 4 : Then we obtain (31) by combining (53) and the above equation. In a similar manner, putting n = 1, n = 2 and n = ne 1 in (43), and from the above equations, we get (30) with the help of (51) and (54). The results (33)–(35) can be derived from (44), (47) and (50) by using Eqs. (30)–(32). By plugging (30)–(32) in (42), (45) and (48), we have:
( 4 X ðk þ hÞ½ðk þ aÞqi kðk þ f þ l lqi qk Þ i
hfqi
i¼1
) a ci ¼ 0;
ð55Þ
4 X ðlqi kÞ½ðk þ aÞqi k ci ¼ 0; 2 fqi i¼1 4 X ci i¼1
qi
ð56Þ
¼ 0:
ð57Þ
with the help of Eqs. (55)–(57), then the value of ci, i = 1, 2, 3, 4 can be determined by using normalization condition. h We now proceed to find the expected net reward of a customer that observes n customers ahead of him and decides to enter. We have the following: Lemma 2. Consider the partially observable M/M/1 queue with breakdowns and delayed repairs where the customers enter to the system according to a threshold strategy ‘While arriving at time t, observe N(t); enter if N(t) 6 ne 1 and balk otherwise’. The net benefit of a customer that observes n customers and decides to enter is given by
! P4 P4 f f 1 1h i¼1 wi qni þ ð1h þ a1Þ i¼1 ci qni ; SðnÞ ¼ R C ðn þ 1Þ 1 þ þ þ P4 n a h l i¼1 ðci þ v i þ wi Þqi 1 Pðne ; rÞ þ 1h þ a1 Pðne ; dÞ f f 1 Sðne Þ ¼ R C ðne þ 1Þ 1 þ þ þ h : a h l Pðne ; rÞ þ Pðne ; sÞ þ Pðne ; dÞ
n ¼ 0; 1; . . . ; ne 1;
ð58Þ ð59Þ
Proof. The expected net reward, if he enters, for a customer that observes n customers is:
SðnÞ ¼ R CTðnÞ;
ð60Þ
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J. Wang, F. Zhang / Applied Mathematics and Computation 218 (2011) 2716–2729
where T(n) = E[SjN = n] denotes his expected mean sojourn time given that he finds n customers in the system just before his arrival. We let Pr (I = ijN = n) be the probability that the state of the server is i when he observes n customers in the system upon his arrival. Conditioning on the state of the server that he finds upon arrival and taking into account (3) and (6) we obtain:
1 1 1 PrðI ¼ djN ¼ nÞ PrðI ¼ rjN ¼ nÞ þ Tðn; sÞ þ þ TðnÞ ¼ Tðn; sÞPrðI ¼ sjN ¼ nÞ þ Tðn; sÞ þ h h a 1 1 1 PrðI ¼ djN ¼ nÞ; ¼ Tðn; sÞ þ PrðI ¼ rjN ¼ nÞ þ þ h h a
ð61Þ
where T(n, 1) is given by (8) and
kPðn; rÞ ; kPðn; rÞ þ kPðn; sÞ þ kPðn; dÞ kPðn; dÞ PrðI ¼ djN ¼ nÞ ¼ ; kPðn; rÞ þ kPðn; sÞ þ kPðn; dÞ
PrðI ¼ rjN ¼ nÞ ¼
n ¼ 0; 1; . . . ; ne :
Using the stationary probabilities obtained in Lemma 1 we obtain the probabilities Pr (I = rjN = n) and Pr (I = djN = n) for n = 0,1, . . . , ne. So, we have:
P4 P4 f f 1 1h i¼1 wi qni þ 1h þ a1 i¼1 ci qni TðnÞ ¼ ðn þ 1Þ 1 þ þ þ ; P4 n a h l i¼1 ðc i þ v i þ wi Þqi 1 Pðne ; rÞ þ 1h þ a1 Pðne ; dÞ f f 1 Tðne Þ ¼ ðne þ 1Þ 1 þ þ þ h ; a h l Pðne ; rÞ þ Pðne ; sÞ þ Pðne ; dÞ
n ¼ 0; 1; . . . ; ne 1;
ð62Þ ð63Þ
where P(ne, r),P(ne, s) and P(ne, d) are given by Eqs. (33)–(35). And then, Eqs. (58) and (59) can be obtained by plugging Eqs. (62) and (63) in (60) for every ne. It should be noted that a customer does not enter the system even if he finds no customers in front of him if S(0) < 0, otherwise, he enters the queue. h Next, we describe the equilibrium balking pure threshold strategies in the partially observable case by assuming S(0) > 0. We have the following result. Theorem 2. Define the sequences (f1(n): n = 0, 1, . . .) and (f2 (n):n = 0, 1, . . .) by
(
) P4 P4 f f 1 1h i¼1 wi qni þ 1h þ a1 i¼1 ci qni ; f1 ðnÞ ¼ R C ðn þ 1Þ 1 þ þ þ P4 n a h l i¼1 ðci þ v i þ wi Þqi
n ¼ 0; 1; . . . ;
ð64Þ
(
f f 1 1 f2 ðnÞ ¼ R C ðn þ 1Þ 1 þ þ þ P4 a h l Wn 1h P4i¼1 v i qn1 þ 1a i¼1 ci qn1 i i !) " # 4 4 4 X fþl f 1 1 X 1 1 1 1X n1 n1 n1 þ ð c þ v þ w Þ q v q þ c q ; þ þ i i i i i i i i h a a i¼1 lh2 la h a i¼1 h2 i¼1
n ¼ 0; 1; . . . ;
ð65Þ
P4 l f 1 n1 where Wn ðfþ lh þ l þ laÞ i¼1 ðc i þ v i þ wi Þqi . By definition, f1(n) = S(n),n = 0,1, . . . , ne 1, f2(ne) = S(ne). Moreover, f1(n) > f2(n), n = 0,1, . . .. Then there exist finite non-negative integers nL 6 nU such that:
f1 ð0Þ; f1 ð1Þ; . . . ; f1 ðnU 1Þ > 0;
f 1 ðnU Þ 6 0;
ð66Þ
and
f2 ðnU Þ; f2 ðnU 1Þ; f2 ðnU 2Þ; . . . ; f2 ðnL Þ 6 0; f2 ðnL 1Þ > 0;
ð67Þ
f2 ðnU Þ; f2 ðnU 1Þ; f2 ðnU 2Þ; . . . ; f2 ð0Þ 6 0:
ð68Þ
or
In the partially observable M/M/1 queue with breakdowns and delayed repairs the pure threshold strategies of the form ‘While arriving at time t, observe N(t); enter if N(t) 6 ne 1 and balk otherwise’, for ne 2 {nL, nL + 1, . . . , nU}, are equilibrium strategies. Proof. We have that f1(0) > 0 if we assume that S(0) > 0 and limn!1 f1 ðnÞ ¼ 1, so if nU is the subscript of the first nonpositive term of the sequence (f1(n)), we have that for the finite number nU the condition (66) holds.
J. Wang, F. Zhang / Applied Mathematics and Computation 218 (2011) 2716–2729
2723
On the other hand, f1(n) > f2(n),n = 0,1, . . . . In particular we conclude that f2(nU) < f1(nU) 6 0. Now, we begin to go backwards, starting from the subscript nU, towards 0 and we let nL 1 be the subscript of the first positive term of the sequence (f2(n)). Then we have (67). If all the terms of (f2(n)) going backwards from nU to 0 are non-positive we have (68). We can now establish the existence of equilibrium threshold policies in the partially observable case. In this model we consider a tagged customer at his arrival instant and assume that all other customers follow the same threshold strategy ‘while arriving at time t, observe N(t), enter if N(t) 6 ne 1 and balk otherwise’ for some fixed ne 2 {nL, nL + 1, . . . , nU}. If the tagged customer finds n 6 ne 1 customers in front of him and decides to enter, his expected benefit is equal to f1(n) > 0 because of (60), (64) and (66). So in this case the customer prefers to enter. If the tagged customer finds n = ne customers in front of him and decides to enter, his expected benefit is equal to f2(ne) 6 0 because of (60), (65), (67) or (68). So in this case the customer prefers to balk. h Define
PðnÞ ¼ Pðn; rÞ þ Pðn; sÞ þ Pðn; dÞ;
n ¼ 1; 2; . . . ; ne :
Because the probability of balking is equal to P(ne), the social benefit per time unit when all customers follow the threshold policies ne given in Lemma 2 equals:
SBpo ¼ kRð1 Pðne ÞÞ C
ne X
nPðnÞ:
ð69Þ
n¼0
Remark 1. Theorem 2 provides an algorithm in order to identify the equilibrium strategies of the partially observable model. One has to start computing (f1(n)) up to the first negative term. This ‘forward’ procedure yields the highest equilibrium threshold nU. Then, one has to start computing (f2(n)) starting from f2(nU) and going towards 0 till the first positive term. This ‘backward’ procedure yields the lowest equilibrium threshold nL. Remark 2. When nL < nU, there exist multiple equilibrium threshold strategies such that ‘While arriving at time t, observe N(t), enter if N(t) 6 ne 2; enter with probability qe if N(t) = ne 1 and balk otherwise’ between every two consecutive equilibrium pure threshold strategies. We have a ‘Follow-The-Crowd’ (FTC) situation when one’s optimal response to a strategy x adopted by all others is increasing in x. In our model, let we consider the function Sne ðnÞ given by (60), for the expected benefit of a tagged customer who decides to enter when he observes n customers in front of him while the others follow the threshold strategy ne. Now suppose that the other customers adopt the strategy ne + 1. Then the tagged customer’s benefit function Sne þ1 ðnÞ coincides with Sne ðnÞ for all n = 0,1, . . . , ne 1 while Sne þ1 ðne Þ ¼ f1 ðne Þ > f2 ðne Þ ¼ Sne ðne Þ. We can then see that a threshold best response of the tagged customer when the others follow the strategy ne + 1 is greater than his best response threshold when the others follow the strategy ne. This means that a tagged customer, whose expected net benefit is given by (60), 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. In a conclusion, the outcome we get in the partially observable case agrees with the results in [7] and [8].
4. Numerical experiments In this section, we show the effect of several parameters on the behavior of customers both in the fully observable and partially observable queueing system with server breakdowns and delayed repairs by some numerical examples. We explore the sensitivity of the equilibrium thresholds and the equilibrium social benefits with respect to some main indicators of the system. In Fig. 3 we observe that thresholds are monotonically decreasing functions of breakdown rate f. This is because frequent breakdowns of the server increase the waiting time of customer and their payment, which makes them reluctant to enter. Figs. 4 and 5 show that if a customer sees the high delayed repair rate a or high repair rate h upon his arrival he prefers to join the queue due to the fact that the period of delayed repair or repair will not last much time. Fig. 6 depicts the equilibrium thresholds versus service rate l. It is intuitive that an arriving customer is more likely to enter when the server can serve more customers per time. Compare Fig. 7 with Fig. 8 we can get some conclusions. The higher reward R the customers gain from service, the higher the waiting cost they can afford. So the thresholds increase with R. However, if waiting costs customers more money, they may be not willing to wait for service avoiding paying too much. Moreover, we can observe that in all first six figures equilibrium thresholds {nL, nL + 1, . . . , nU} for the partially observable case always locate between ne(d) and ne(s) for the fully observable case. In other words, the thresholds in the partially model have intermediate values between the two separate thresholds provided that the state of the server can be observed by arriving customers. Figs. 9–12 are concerned with the social benefit under the equilibrium strategy for the different information levels. For the partially observable case we present the social benefit under the two extreme thresholds nL and nU. In regard to the sensitivity of social benefit in equilibrium, we observe that it increases with respect to a, h and R, whereas decreases with respect to the breakdown rate f, which are intuitive. Furthermore, the four figures show that the social benefit of all customers
2724
J. Wang, F. Zhang / Applied Mathematics and Computation 218 (2011) 2716–2729 22 n (r) e
20
n (s) e
n (d) e
18
n
L
n
U
16
Thresholds
14
12
10
8
6
4
2 0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0.45
0.5
ζ Fig. 3. Thresholds vs. f for k = 0.5, l = 1, a = 0.3, h = 0.2, C = 1, R = 30.
18 n (r) e
16
n (s) e
n (d) e
n
14
L
n
U
Thresholds
12
10
8
6
4
2
0 0.05
0.1
0.15
0.2
0.25
α
0.3
0.35
0.4
Fig. 4. Thresholds vs. a for k = 0.5, l = 1, f = 0.1, h = 0.2, C = 1, R = 30.
2725
J. Wang, F. Zhang / Applied Mathematics and Computation 218 (2011) 2716–2729
20 n (r) e
18
n (s) e
n (d) e
16
n
L
n
U
14
Thresholds
12
10
8
6
4
2
0 0.05
0.1
0.15
0.2
0.25
0.3
θ
0.35
0.4
0.45
0.5
Fig. 5. Thresholds vs. h for k = 0.5, l = 1, a = 0.3, f = 0.1, C = 1, R = 30.
90 n (r) e
80
n (s) e
n (d) e
70
n
L
n
U
Thresholds
60
50
40
30
20
10
0 0.5
1
1.5
2
2.5
3
3.5
4
μ Fig. 6. Thresholds vs. l for k = 0.5, a = 0.3, f = 0.1, h = 0.2, C = 1, R = 30.
4.5
5
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J. Wang, F. Zhang / Applied Mathematics and Computation 218 (2011) 2716–2729
20 n (r) e
n (s) e
18
n (d) e
n
L
n
Thresholds
16
U
14
12
10
8
6 20
22
24
26
28
30
32
34
36
38
R Fig. 7. Thresholds vs. R for k = 0.5, l = 1, a = 0.3, f = 0.1, h = 0.2, C = 1.
16 n (r) e
n (s) e
14
n (d) e
n
L
n
Thresholds
12
U
10
8
6
4
2
1
1.1
1.2
1.3
1.4
1.5
1.6
1.7
C Fig. 8. Thresholds vs. C for k = 0.5, l = 1, a = 0.3, f = 0.1, h = 0.2, R = 30.
1.8
1.9
2727
J. Wang, F. Zhang / Applied Mathematics and Computation 218 (2011) 2716–2729
11 P.O.nL
10
P.O.nU
9
F.O.
Equilibrium social benefit
8
7
6
5
4
3
2
1 0.05
0.1
0.15
0.2
0.25
ζ
0.3
0.35
0.4
0.45
0.5
Fig. 9. Equilibrium social benefit for different information levels vs. f for k = 0.5, l = 1, a = 0.3, h = 0.2, C = 1, R = 30.
9
8
P.O.n
L
P.O.n
U
7
Equilibrium social benefit
F.O.
6
5
4
3
2
1 0.05
0.1
0.15
0.2
0.25
α
0.3
0.35
0.4
0.45
0.5
Fig. 10. Equilibrium social benefit for different information levels vs. a for k = 0.5, l = 1, f = 0.1, h = 0.2, C = 1, R = 30.
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J. Wang, F. Zhang / Applied Mathematics and Computation 218 (2011) 2716–2729
10
P.O.n
L
9
P.O.n
U
Equilibrium social benefit
8
F.O.
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Fig. 11. Equilibrium social benefit for different information levels vs. h for k = 0.5, l = 1, a = 0.3, f = 0.1, C = 1, R = 30.
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in the fully observable case is not always more than that in the partially observable case. This reveals that the information about the system is not always helpful for increasing social benefit in equilibrium. That is, in some cases more accurate information improves the equilibrium social benefit. However, in other cases information can actually hurt the customers. 5. Conclusion Inspired by Economou and Kanta [2], we studied equilibrium balking strategies of the observable single-server queue with breakdowns and delayed repairs. The equilibrium threshold balking strategies and equilibrium social benefits for all customers are derived respectively for the fully and partially observable systems with server breakdowns and delayed repairs. It is the first time that server’s delayed repair process is taken into account for the effect of system information on the strategic behavior in queueing systems. Algorithms are also provided to identify the equilibrium strategies for the fully observable system and partially observable system. We observe by numerical examples that the thresholds in the partially observable system take intermediate values between two extremes, i.e., two separate thresholds in the fully observable case. ‘‘Follow-The-Crowd’’ (FTC) property has also been observed. Moreover, it is shown that the information about the system is not always helpful for increasing social benefit in equilibrium. In some cases more accurate information improves the equilibrium social benefit, but in other cases can hurt the customers. A new property in this model is that the repair time has two stages and hence it is not memoryless. So, the analysis on the Nash equilibria of the present model with delayed repairs is more general and practical than the model with immediate repair. Acknowledgements The authors thank the anonymous referee’s comments leading to a much improved presentation. This work was supported by the National Natural Science Foundation of China (Nos. 10871020 and 11010301053). References [1] B. Avi-Itzhak, P. Naor, Some queueing problems with the service station subject to breakdown, Operations Research 10 (1962) 303–320. [2] A. Economou, S. Kanta, Equilibrium balking strategies in the observable single-server queue with breakdowns and repairs, Operations Research Letters 36 (2008) 696–699. [3] N.M. Edelson, K. Hildebrand, Congestion tolls for Poisson queueing processes, Econometrica 43 (1975) 81–92. [4] P.F. Guo, P. Zipkin, Analysis and comparison of queues with different levels of delay information, Management Science 53 (2007) 962–970. [5] R. Hassin, Consumer information in markets with random product quality: the case of queues and balking, Econometrica 54 (1986) 1185–1195. [6] R. Hassin, Information and uncertainty in a queueing system, Probablity in the Engineering and Informational Sciences 21 (2007) 361–380. [7] R. Hassin, M. Haviv, Equilibrium threshold strategies: the case of queues with priorities, Operations Research 45 (1997) 966–973. [8] R. Hassin, M. Haviv, To Queue or Not to Queue: Equilibrium Behavior in Queueing Systems, Kluwer Academic Publishers, Boston, 2003. [9] R. Hassin, M. Henig, Control of arrivals and departures in a state-dependent input–output system, Operations Research Letters 5 (1986) 33–36. [10] P. Naor, The regulation of queue size by levying tolls, Econometrica 37 (1969) 15–24. [11] S. Stidham Jr., Optimal Design of Queueing Systems, CRC Press, Taylor and Francis Group, Boca Raton, 2009. [12] J. Wang, J. Cao, Q. Li, Reliability analysis of the retrial queue with server breakdowns and repairs, Queueing Systems 38 (2001) 363–380.