Equilibrium customer strategies in Markovian queues with partial breakdowns

Computers & Industrial Engineering 66 (2013) 751–757

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Equilibrium customer strategies in Markovian queues with partial breakdowns Le Li, Jinting Wang ⇑, Feng Zhang Department of Mathematics, Beijing Jiaotong University, Beijing 100044, China

a r t i c l e

i n f o

Article history: Received 4 December 2012 Received in revised form 14 September 2013 Accepted 24 September 2013 Available online 3 October 2013 Keywords: M/M/1 queue Nash equilibrium strategies Balking Partial breakdowns

a b s t r a c t We consider equilibrium analysis of a single-server Markovian queueing system with working breakdowns. The system may become defective at any point of time when it is in operation. However, when the system is defective, instead of stopping service completely, the service continues at a slower rate. We assume that the arriving customers decide whether to join the system or balk based on a natural reward-cost structure. With considering waiting cost and reward, the balking behavior of customers is investigated and the corresponding Nash equilibrium strategies are derived. The effects of the information level on the equilibrium behavior are illustrated further via numerical experiments. Ó 2013 Elsevier Ltd. All rights reserved.

1. Introduction Recently, the economic analysis of customer behavior on the performance of a queueing system has been studied extensively. Early works on the M/M/1 model with a reward-cost structure include Naor (1969) and Edelson and Hildebrand (1975). Naor studied the M/M/1 model in which an arriving customer observes the number of customers and then makes his decision whether to join or balk in observable case. Edelson and Hildebrand considered the corresponding unobservable case in the above model. Hassin and Haviv (1997) dealt with equilibrium threshold strategies in queues with priorities. The monograph of Hassin and Haviv (2003) summarized the main methodologies and results in the area of the game-theoretical economic analysis of queueing systems. Nowadays, there is an increasing number of papers that deal with the economic analysis of the balking behavior of customers in variants of the M/M/1 queue, see e.g. Burnetas and Economou (2007) (M/M/ 1 queue with setup times), Sun, Guo, and Tian (2010) (M/M/1 queue with setup/closedown times), Economou and Manou (2012) (a clearing queueing system in alternating environment), Guo and Hassin (2011) (strategic behavior and social optimization in Markovian vacation queues), Economou and Kanta (2011) (the single-server constant retrial queue), Boudali and Economou (2012) (the single server Markovian queue with catastrophes), among others.

⇑ Corresponding author. Tel.: +86 1051688449; fax: +86 10 5184 0433. E-mail address: [email protected] (J. Wang). 0360-8352/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.cie.2013.09.023

In the queueing literature most of papers assume that the server is always available, although this assumption is evidently unrealistic. Actually, perfectly reliable servers are virtually nonexistent. The servers in many practical service systems may well be subject to lengthy and unpredictable breakdowns while serving a customer. For example, in manufacturing systems the machine may break down due to failures of machines or job related problems. Economic analysis of customer behavior on the performance of such kind of queueing systems has also been carried out by several authors. Economou and Kanta (2008) studied the equilibrium balking strategies for customers in the observable M/M/1 queue with an unreliable server and repairs. Later on, Wang and Zhang (2011) extended this model to the delayed repairs situation. They observed that 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 need a setup time (delayed repair time). The equilibrium threshold balking strategies and the equilibrium social benefit for all customers are derived for the fully and partially observable system respectively. It is common in the above studies to assume that the servers will stop service completely when the system breaks down due to various factors. However, due to complexity of modern operational systems, a multi-element system may degrade in a multistate failure mode other than the traditional two-state (working or failure) process. There are many situations in the real world where the breakdown of a server may not stop the service of a customer completely. Gnedenko and Kovalenko (1989) called systems with partial failure early and discussed the influence of partial failure in M/G/1 system with an unreliable and ‘‘renewable’’ server.

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Later on, Sridharan and Jayashree (1996) studied some characteristics of a finite queue with normal, partial and total failures. BaykalGursoy and Xiao (2004) investigated an M/M/1 queue with Markov modulated service rates. More recently, Kalidass and Kasturi (2012) introduced the concept of working breakdowns in their study. It is assumed that when the servers are failed the service continues at a slower rate instead of stopping service completely. Such a problem arises from computer-based technology and related application area including telecommunication, flexible manufacturing, e-commerce and supply-chain systems. It should be pointed out that the concept of partial breakdown is different from working breakdown. In a system with partial breakdowns and repairs, the failure of a server does not stop the service of customers completely, but serves customers at a slower rate than normal working rate. The partial breakdowns can occur at any state no matter whether the server is busy or idle, while in the system with working breakdowns, the server may break down only when it is in the operational state. In a word, when a system is idle, the server may fail in the system with partial breakdowns but cannot fail in the system with working breakdowns. It is worthwhile to point out that there are some recent researches which investigate the models with variable service rates from different perspectives, such as working vacation models or queueing models with slower servers. Servi and Finn (2002) firstly formulated queueing models with working vacations and obtained the steady-state distribution for the number of customers and the expected sojourn time in M/M/1 queue. Tian, Zhao, and Wang (2008) studied the M/M/1 queue with single working vacation and gave some further results. Zhang, Wang, and Liu (2013) analyzed equilibrium strategies in Markovian queues with working vacations and obtained explicit optimal strategies under different information levels. The interested readers are referred to Zhang and Hou (2010) and Jain and Upadhyaya (2011) and references therein for the recent advances. However, there are some differences between working vacations and partial breakdowns mentioned above. More specifically, a working vacation must occur only after completing the service of the customers in the system, but the partial breakdowns can occur at the any point of time no matter the server is busy or idle. So the concept of partial breakdowns differs from the concept of working vacation. In addition, when the server is at partial breakdowns state, it works at a rate which is slower than normal rate, just as the slow server studied in Perel and Yechiali (2009). It is also found that the models with slow server and with partial breakdowns are same intrinsically, for example, see Perel and Yechiali (2009) and Sridharan and Jayashree (1996). The present paper aims to study the equilibrium behavior of the customers in the framework of an M/M/1 queueing model with partial breakdowns. The balking behavior of customers in practical service systems has important effect on the performance and related economic activities. The behavior of customers under different conditions of the system should be taken into account to obtain a reliable representation of what is going on in these systems. It is meaningful to determine equilibrium balking strategies for the customers when the breakdown of a server may not stop the service of a customer completely. In this work we will carry out the equilibrium analysis of the system under the conditions that the system is fully observable and fully unobservable. The equilibrium customer strategies in the two cases will be investigated extensively. The remaining part of the paper is organized as follow. In Section 2, we introduce the model and the reward-cost structure. In Section 3, we consider the equilibrium strategies for fully observable queues. In Section 4, we consider the equilibrium strategies for fully unobservable queues. In Section 5, some numerical examples are presented to illustrate the effects of several parameters on the

customers’ behavior in the considered models. Finally, in Section 6, some conclusions are given.

2. Description of the model We consider an M/M/1 queueing system with an infinite waiting room where customers arrive according to a Poisson process with rate k. We assume the service alternates between two states with state space I = {1, 0} that are exponentially distributed at rates f and h respectively. When the server is at state 1, customers are served at a rate of l. As natural factor or human factor influences the rate of service, the customers are served at a lower rate of l0 with partial breakdowns state 0. We assume the service times are both exponentially distributed and l0 < l. We represent the state of the station at time t by the pair (N(t), I(t)), where N(t) and I(t) denote the number of customers and the state of the server (1: normal working state; 0: partial breakdowns state). It is clear that the process {N(t), I(t):t P 0} is a continuous-time Markov chain with non-zero transition rates by

qðn;iÞðnþ1;iÞ ¼ k;

n ¼ 0; 1; 2; . . . ; i ¼ 0; 1;

qðn;1Þðn1;1Þ ¼ l; n ¼ 1; 2 . . . ; qðn;0Þðn1;0Þ ¼ l0 ; n ¼ 1; 2; . . . ; qðn;0Þðn;1Þ ¼ h;

n ¼ 0; 1; 2; . . . ;

qðn;1Þðn;0Þ ¼ f;

n ¼ 0; 1; 2; . . .

:

The corresponding transition rate diagram is shown in Fig. 1. We are interested in the behavior of customers when they are allowed to decide whether to join or balk. We assume every customer receives a reward of R utility units after service. This may reflect his satisfaction or the added value of being served. In addition, there exists a waiting cost of C utility units per time unit when a customer remains in the system including the time of waiting in the queue and being served. We also assume that 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. In the next section we obtain customer optimal strategies for joining/balking. Customers are aware of the rates of transition from normal to partial breakdowns state upon arrival and vise versa. Besides, we consider two cases depending on the levels of system information available to customers at their arrival instants, before the decision is made:  Fully observable case: customers observe the queue length N(t) and the server state I(t).  Fully unobservable case: customers do not observe the queue length N(t) or the server state I(t). 3. Equilibrium threshold strategies for fully observable case In this section, we show that there exist equilibrium strategies of threshold type in the fully observable case. In the fully observable case where customers are informed both the number of customers present and the state of the server upon arrival, a pure threshold strategy is specified by a pair (ne(0), ne(1)) and the

Fig. 1. Transition rate diagram of the original model.

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L. Li et al. / Computers & Industrial Engineering 66 (2013) 751–757

balking strategy has the form ‘while arriving at time t, observe (N(t), I(t)); enter if N(t) 6 ne(I(t)) and balk otherwise’. We thus conclude the following results. Theorem3.1. In the fully observable M/M/1 queue with partial breakdowns, there exist a pair of thresholds (ne(0), ne(1)), such that the strategy ‘observe (N(t), I(t)) upon arrival, enter if N(t) 6 ne(I(t)) and balk otherwise’, and (ne(0), ne(1)) = ðbx0 c; bx1 cÞ, where xi is the unique root of equation

ax þ bi c

xþ1

þ di ¼ 0;

i ¼ 0; 1;

ð3:1Þ

Tðn;0Þ  Tðn;1Þ ¼ n ¼ 1;2;. ..;





b1 ¼ 

ð3:3Þ

lhðl  l0 ÞC b0 ¼ ; ðlh þ l0 fÞ2 c¼

ð3:4Þ

d1 ¼ R 

ð3:5Þ !

C

ll0 þ l0 f þ lh

d0 ¼ d1 

l0 þ f þ h 

ll20 fðl  l0 Þ ; ðlh þ l0 fÞ2

Cðl  l0 Þ ; lh þ l0 f

ð3:6Þ

ð3:7Þ

and S(n, i) = an + bicn+1 + di is a monotone decreasing function of n. Proof. It is obvious that for an arriving customer based on the reward-cost structure, his expected net reward if he enters is

Sðn; iÞ ¼ R  CTðn; iÞ;

ð3:8Þ

where T(n, i) denotes his expected sojourn time given that he finds the system at state (n, i) upon his arrival. Then we have the following system of equations:

Tðn; 0Þ ¼

1 l0 h þ Tðn  1; 0Þ þ Tðn; 1Þ; l0 þ h l0 þ h l0 þ h

n ¼ 1; 2; . . . ; Tð0; 0Þ ¼

Tð0; 1Þ ¼

Tðn; 1Þ ¼

1

ð3:9Þ h

þ

l0 þ h l0 þ h 1

f

þ

lþf lþf

Tð0; 1Þ;

Tð0; 0Þ;

ð3:10Þ

ð3:11Þ

1 l f þ Tðn  1; 1Þ þ Tðn; 0Þ; lþf lþf lþf

n ¼ 1; 2; . . . :

ð3:12Þ

Solving the system of (3.10) and (3.11) we obtain T(0, 0) and T(0, 1). Easily we can find that

Tð0; 0Þ  Tð0; 1Þ ¼

l  l0 ; ll0 þ l0 f þ lh

ð3:13Þ

From Eqs. (3.12) and (3.9) we can obtain:

1

ðl þ f þ h þ lðl0 þ hÞ ll0 þ l0 f þ lh 0  Tðn  1; 1Þ þ l0 fTðn  1; 0ÞÞ;

Tðn; 1Þ ¼

ð3:14Þ

ð3:16Þ

By plugging (3.16) in (3.14) we have

l0 þ f þ h þ Tðn  1; 1Þ ll0 þ l0 f þ lh   n  l0 fðl  l0 Þ ll0 ; þ 1 ll0 þ l0 f þ lh ðl0 f þ lhÞðll0 þ l0 f þ lhÞ

n ¼ 1; 2; . . . ;

ð3:17Þ

We can iterate (3.17) to compute T(n, 1) and then we use again (3.16) to obtain T(n, 0). We have

Tðn; 1Þ ¼

ll0 ; ll0 þ l0 f þ lh

;

n ¼ 0; 1; 2; . . . ;

ð3:2Þ

l0 fðl  l0 ÞC ; ðlh þ l0 fÞ2

nþ1 !

l  l0 ll0 Tðn; 0Þ  Tðn; 1Þ ¼ 1 l0 f þ lh ll0 þ l0 f þ lh

Tðn; 1Þ ¼



l fðl  l0 Þ ; a¼ l þfþhþ 0 ll0 þ l0 f þ lh 0 lh þ l0 f

ð3:15Þ

We can use (3.13) and iterate (3.15) to obtain

where

C

l  l0 ll0 þ ðTðn  1;0Þ  Tðn  1;1ÞÞ; ll0 þ l0 f þ lh ll0 þ l0 f þ lh





l0 þ f þ h l0 fðl  l0 Þ þ n ll0 þ l0 f þ lh ðl0 f þ lhÞðll0 þ l0 f þ lhÞ  nþ1 l fðl  l0 Þ ll0 þ 0 2 ðl0 f þ lhÞ ll0 þ l0 f þ lh þ

 Tðn; 0Þ ¼

1

ll0 þ l0 f þ lh

!

l0 þ f þ h 

ll20 fðl  l0 Þ ; ðlh þ l0 fÞ2

ð3:18Þ



l0 þ f þ h l0 fðl  l0 Þ n þ ll0 þ l0 f þ lh ðl0 f þ lhÞðll0 þ l0 f þ lhÞ ! nþ1 lhðl  l0 Þ ll0  2 ll0 þ l0 f þ lh ðlh þ l0 fÞ ! 1 ll20 fðl  l0 Þ þ l þfþh ll0 þ l0 f þ lh 0 ðlh þ l0 fÞ2 l  l0 : 8n 2 N; ð3:19Þ þ lh þ l0 f

1 Sðn; 1Þ  Sðn  1; 1Þ ¼ C ll0 þ l0 f þ lh   n  l fðl  l0 Þ l0 fðl  l0 Þ ll0 :  l0 þ f þ h þ 0  lh þ l0 f lh þ l0 f ll0 þ l0 f þ lh ð3:20Þ 0 Since 0 < ll þll < 1, S(n, 1)  S(n  1, 1) is decreasing in n. 0 l0 fþlh When n = 1, S(1, 1)  S(0, 1) < 0, so S(n, 1)  S(n  1, 1) < 0 for n = 1, 2, 3. . .. Similarly we can show that S(n, 0)  S(n  1, 0) < 0, so S(n, i), i = 0, 1 is a monotone decreasing function. h

A customer does not enter the system if S(n, i) < 0, otherwise, he enters the queue. We can use (3.8), (3.18), and (3.19) to obtain that the customer arriving at time t decides to enter if and only if n 6 ne(i), where (ne(0), ne(1)) are obtained by using the unique solution xi of equation (3.1).

4. Equilibrium mixed strategy for fully unobservable case Now we turn our attention to the unobservable queue where customers do not observe the state of the system at all. In the fully unobservable case, a mixed strategy has the form ‘‘while arriving at time t, do not observe N(t) and I(t), enter with probability q.’’ The corresponding transition diagram is shown in Fig. 2.

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L. Li et al. / Computers & Industrial Engineering 66 (2013) 751–757

Hence, using Little’s law, the expected sojourn time if a customer decides to enter upon his arrival is

E½W ¼ Fig. 2. Transition rate diagram for the q mixed strategy.

¼

Lemma 4.1. In the fully unobservable model of M/M/1 queue with partial breakdowns and k < l0 , his expected sojourn time if a customer decides to enter is

E½W ¼

ðh þ fÞ2 ðuðg 1 Þ þ hg 1 Þ þ hðl  l0 Þð1  g 1 Þðfg 1  uðg 1 ÞÞ : ðh þ fÞðuðg 1 Þ þ hg 1 Þ½fðl0  kqÞ þ hðl  kqÞ

EðNÞ G0 ð1Þ ¼ kq kq ðh þ fÞ2 ðuðg 1 Þ þ hg 1 Þ þ hðl  l0 Þð1  g 1 Þðfg 1  uðg 1 ÞÞ : ðh þ fÞðuðg 1 Þ þ hg 1 Þ½fðl0  kqÞ þ hðl  kqÞ

The social benefit per time unit when all customers follow a mixed policy qe is obtained as 2

SBfu ¼ kqe R 

Ckq½ðh þ fÞ ðuðg 1 Þ þ hg 1 Þ þ hðl  l0 Þð1  g 1 Þðfg 1  uðg 1 ÞÞ : ðh þ fÞðuðg 1 Þ þ hg 1 Þ½fðl0  kqe Þ þ hðl  kqe Þ

ð4:1Þ Proof. The proof is similar to Section 4 in Kalidass and Kasturi (2012). The steady state equations governing the fully unobservable queue are given by

ðkq þ fÞpð0; 1Þ ¼ lpð1; 1Þ þ hpð0; 0Þ; ðkq þ f þ lÞpðn; 1Þ ¼ kqpðn  1; 1Þ þ hpðn; 0Þ þ lpðn þ 1; 1Þ;

ðkq þ hÞpð0; 0Þ ¼ fpð0; 1Þ þ l0 pð1; 0Þ; ðkq þ h þ l0 Þpðn; 0Þ ¼ kqpðn  1; 0Þ þ fpðn; 1Þ þ l0 pðn þ 1; 0Þ;

ð4:2Þ

pðn; 1Þzn ;

G0 ðzÞ ¼

n¼0

Lemma 4.2. In the fully unobservable model of M/M/1 queue with partial breakdowns and k < l0 , if a customer decides to enter, his expected sojourn time E[W] is strictly increasing for q  [0, 1].

ð4:3Þ

Proof. We let

ð4:4Þ

/1 ðqÞ ¼ kg 21 h2 ðl  l0 Þð1  g 1 Þ2 ½fðl0  kqÞ þ hðl  kqÞ;

n  1:

We define the probability generating functions as follows:

G1 ðzÞ ¼

h

n  1;

ð4:5Þ 1 X

ð4:11Þ

1 X

/2 ðqÞ ¼ khfg 21 ðl  l0 Þð1  g 1 Þ2 ½fðl0  kqÞ þ hðl  kqÞ; /3 ðqÞ ¼kðh þ fÞ3 ðuðg 1 Þ þ hg 1 Þ2 þ khfg 1 ðl  l0 Þð1  g 1 Þðh þ fÞ  ðuðg 1 Þ þ hg 1 Þ

pðn; 0Þzn ;

n¼0

khuðg 1 Þðh þ fÞðl  l0 Þð1  g 1 Þðuðg 1 Þ þ hg 1 Þ:

From Eqs. (4.1)–(4.5), we get

G0 ðzÞ ¼

fkqzð1  zÞ þ fz þ lðz  1Þgl0 pð0; 0Þ þ lfzpð0;1Þ fkqzð1  zÞ þ hz þ l0 ðz  1Þgfkqzð1  zÞ þ fz þ lðz  1Þg  hfz2 ðz  1Þ;

So

dE½W ¼ dq

ð4:6Þ ¼

fkqzð1  zÞ þ hz þ l0 ðz  1Þglpð0; 1Þ þ l0 hzpð0; 0Þ G1 ðzÞ ¼ fkqzð1  zÞ þ hz þ l0 ðz  1Þgfkqzð1  zÞ þ fz þ lðz  1Þg  hfz2 ðz  1Þ: ð4:7Þ Denominator of G1(z) can be rewritten as (1  z)U(z), where UðzÞ ¼ ðl  kqzÞðl0  kqzÞð1  zÞ  fzðl0  kqzÞ  hzðl  kqzÞ. l l As the three roots of U(z) lie in (0, 1), (1,kq ) and (kq ; þ1), so only one root of U(z) lies between 0 and 1. Let the root be g1. In addition

uðzÞ ¼ ðl0  kqzÞð1  zÞ  hz, has two roots z1 < 1, z2 > 1. From u(0) > 0, G1(z) P 0 for 0 6 z 6 1, the numerator of G1(z) must vanish at z = g1. From (4.7) we have

pð0; 0Þ ¼

luðg 1 Þ pð0; 1Þ: hl0 g 1 hg 1 ½fðl0  kqÞ þ hðl  kqÞ : ðh þ fÞðluðg 1 Þ þ hlg 1 Þ

ð4:9Þ

Therefore, we can have

GðzÞ ¼ G0 ðzÞ þ G1 ðzÞ ½uðg 1 Þ þ hg 1 ½kqzð1  zÞ þ ðf þ hÞz þ ðz  1Þ½luðg 1 Þ þ hl0 g 1  UðzÞ fðl0  kqÞ þ hðl  kqÞ : ð4:10Þ  ðh þ fÞðuðg 1 Þ þ hg 1 Þ ¼

/1 ðqÞ þ /2 ðqÞ þ /3 ðqÞ ðh þ fÞðuðg 1 Þ þ hg 1 Þ2 ½fðl0  kqÞ þ hðl  kqÞ2

;

Considering the conditions l0 > kq, l > l0, q e [0, 1] and 0 < g1 < 1, we obtain that /1(q) > 0, /2(q) > 0. We know that E[W] is a non-negative value, hence

ðh þ fÞ2 ðuðg 1 Þ þ hg 1 Þ þ hðl  l0 Þð1  g 1 Þðfg 1  uðg 1 ÞÞ  0; which leads to that /3(q) P 0. Thus, we conclude that E[W] is strictly increasing for q e [0, 1]. h We can describe the equilibrium behavior of the customers in the following.

ð4:8Þ

Using the normalizing condition G1(1) + G0(1) = 1, we obtain

pð0; 1Þ ¼

( )0 ðh þ fÞ2 ðuðg 1 Þ þ hg 1 Þ þ hðl  l0 Þð1  g 1 Þðfg 1  uðg 1 ÞÞ ðh þ fÞðuðg 1 Þ þ hg 1 Þ½fðl0  kqÞ þ hðl  kqÞ

Theorem 4.1. In the fully unobservable model of M/M/1 queue with partial breakdowns and k < l0 , a unique mixed Nash equilibrium strategy ‘enter with probability qe’ exists, where qe is given by

8 R > < 0; 0 < C  Tð0Þ;  qe ¼ qe ; Tð0Þ < CR < Tð1Þ; > : 1; CR  Tð1Þ;

ð4:12Þ

where qe is the unique root of equation for q e [0, 1]

R

C½ðh þ fÞ2 ðuðg 1 Þ þ hg 1 Þ þ hðl  l0 Þð1  g 1 Þðfg 1  uðg 1 ÞÞ ¼ 0; ðh þ fÞðuðg 1 Þ þ hg 1 Þ½fðl0  kqÞ þ hðl  kqÞ

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L. Li et al. / Computers & Industrial Engineering 66 (2013) 751–757

and

26 n (1) e

2

hþf hfðl  l0 Þ Tð0Þ ¼ þ ; fl0 þ hl ðh þ fÞðfl0 þ hlÞðll0 þ fl0 þ hlÞ

ð4:13Þ

n (0) e

24

ðh þ fÞ2 ðl0  kg 11 Þ þ hðl  l0 Þ½ðh þ fÞg 11  ðl0  kg 11 Þð1  g 11 Þ ; ðh þ fÞðl0  kg 11 Þ½fðl0  kÞ þ hðl  kÞ ð4:14Þ

where g11 is the unique root of U(z) between 0 and 1 when q = 1.

23

Thresholds

Tð1Þ ¼

25

22 21 20

Proof. If a tagged customer at his arrival instant decides to enter, his expected net benefit is

19 18

C½ðh þ fÞ2 ðuðg 1 Þ þ hg 1 Þ þ hðl  l0 Þð1  g 1 Þðfg 1  uðg 1 ÞÞ Sfu ðqÞ ¼ R  ; ðh þ fÞðuðg 1 Þ þ hg 1 Þ½fðl0  kqÞ þ hðl  kqÞ

17 0.05

0.1

0.15

0.2

0.25

ð4:15Þ and we have ( Sfu ð0Þ ¼ R  CTð0Þ ¼ R  C

2

)

0.4

0.45

0.5

26

ð4:16Þ

n (1) e

25

Sfu ð1Þ ¼R  CTð1Þ ¼ R  C ( ) ðh þ fÞ2 ðl0  kg 11 Þ þ hðl  l0 Þ½ðh þ fÞg 11  ðl0  kg 11 Þð1  g 11 Þ ;  ðh þ fÞðl0  kg 11 Þ½fðl0  kÞ þ hðl  kÞ

n (0) e

24 23

Thresholds

where g11 is the unique root of U(z) between 0 and 1 when q = 1. h

0.35

Fig. 3. Thresholds vs. f for k ¼ 0:5; l ¼ 1; l0 ¼ 0:5; h ¼ 0:2; C ¼ 1; R ¼ 30:

hþf hfðl  l0 Þ þ ; fl0 þ hl ðh þ fÞðfl0 þ hlÞðll0 þ fl0 þ hlÞ

ð4:17Þ

0.3

ζ

22 21

From Lemma 4.2, we compute that E[W] is strictly increasing for q e [0, 1]. Therefore,

20

 When 0 < RC  Tð0Þ, Sfu(q) is non-positive for every q, the best response of a customer upon arrival is balking and the unique equilibrium point is q = 0, which gives the first branch of (4.12).  When CR 2 ðTð0Þ; Tð1ÞÞ, there exists a unique solution qe of the equation Sfu(q) = 0 in the interval (0, 1), thus we have the second branch of (4.12).  When RC  Tð1Þ, Sfu(q) is positive for every q, thus the best response is q = 1. So entering is the unique Nash equilibrium strategy and we have the third branch of (4.12).

18

19

17 0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

θ Fig. 4. Thresholds vs. h for k ¼ 0:5; l ¼ 1; l0 ¼ 0:5; f ¼ 0:1; C ¼ 1; R ¼ 30:

42 n (1) e

40

5. Numerical examples

n (0) e

38 36

Thresholds

In this section, we study the effect of the information levels as well as several parameters on the behavior of customers in the fully observable and fully unobservable case. Specifically, we explore the sensitivity of the equilibrium thresholds for the fully observable model. We also study the sensitivity of the expected sojourn time when a customer upon arrival decides to enter. In Fig. 3 we observe that thresholds are monotonically decreasing functions of transition rate f. This is because the customers are served at a lower service rate so as to increase their overall delay, when the server converts to the partial breakdowns state from normal working state more frequently. Therefore, they enter reluctantly. Fig. 4 shows customers have a greater incentive to enter if the server has more normal working periods. Figs. 5 and 6 depict the equilibrium thresholds vs. service rates l and l0. It is intuitive that an arriving customer is more likely to enter when the server can serve more customers per time unit. Comparing Figs. 7 and 8 we know that 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,

34 32 30 28 26 24 22

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

μ Fig. 5. Thresholds vs. l for k ¼ 0:5; l0 ¼ 0:5; h ¼ 0:2; f ¼ 0:1; C ¼ 1; R ¼ 30:

they may be not willing to wait for service avoiding paying too much. Fig. 9 shows the expected sojourn time of a customer who

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L. Li et al. / Computers & Industrial Engineering 66 (2013) 751–757

3.5

24 n (1) e

23

n (0) e

3

21

2.5

E [W]

Thresholds

22

20

2

19 18

1.5 17 16 0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

μ0

1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

q

Fig. 6. Thresholds vs. l0 for k ¼ 0:5; l ¼ 1; h ¼ 0:2; f ¼ 0:1; C ¼ 1; R ¼ 30:

Fig. 9. E[W] vs. q for k ¼ 0:5; l ¼ 1; l0 ¼ 0:5; h ¼ 0:2; f ¼ 0:1.

interval [0, 1], the number of customers arriving and joining the system increases.

24 n (1) e

n (0)

22

e

6. Conclusions and extensions

Thresholds

20 18 16 14 12 10

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

C Fig. 7. Thresholds vs. C for k ¼ 0:5; l ¼ 1; l0 ¼ 0:5; h ¼ 0:2; f ¼ 0:1; R ¼ 30:

30

Acknowledgments

n (1) e

28

n (0) e

This work was supported by the National Natural Science Foundation of China (Grant No. 11171019), Program for New Century Excellent Talents in University (No. NCET-11-0568) and the Fundamental Research Funds for the Central Universities (Nos. 2011JBZ012 and 2013JBZ019).

26

Thresholds

In this paper we considered the problem of analyzing customer strategic behavior, in the fully observable and fully unobservable M/M/1 queues with partial breakdowns, where customers decide whether to join the queue or balk upon arrival. We study the equilibrium strategies in the two cases. We also discussed the sensitivity of the equilibrium thresholds with respect to various parameters and the sensitivity of the expected sojourn time when a customer upon arrival decides to enter. This work can be extended in several directions. One direction is to study the equilibrium strategies in a system where customers have access to partial information, i.e., customers are only informed about the queue length N(t) or the server state I(t) upon arrival. For another, we can discuss the social benefit for different information levels. Some preliminary work has shown that the equilibrium analysis in these problems is more complicated, and non-threshold equilibrium may exist.

24 22

References

20 18 16 14 20

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24

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30

32

34

36

38

R Fig. 8. Thresholds vs. R for k ¼ 0:5; l ¼ 1; l0 ¼ 0:5; h ¼ 0:2; f ¼ 0:1; C ¼ 1:

decides to enter is monotonically increasing function of probability q in the unobservable case. Because with increasing of q in the

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