Equilibrium balking behavior in the Geo=Geo=1 queueing system with multiple vacations

Equilibrium balking behavior in the Geo=Geo=1 queueing system with multiple vacations

Yan Ma

^b,a

, Wei-qi Liu

^b,c

, Ji-hong Li

^c,⇑

aSchool of Mathematics and Statistics, Central South University, Changsha 410075, China

bCollege of Mathematical Sciences, Shanxi University, Taiyuan 030006, China

cCollege of Management, Shanxi University, Taiyuan 030006, China

a r t i c l e i n f o

Article history:

Received 12 November 2011 Received in revised form 7 July 2012 Accepted 16 August 2012

Available online 4 September 2012

Keywords:

Economics of queues Multiple vacations

Equilibrium balking strategies Stationary distribution Social benefit

Matrix-geometric solution method

a b s t r a c t

This paper studies the equilibrium behavior of customers in theGeo=Geo=1 queueing sys- tem under multiple vacation policy. The server leaves for repeated vacations as soon as the system becomes empty. Customers decide for themselves whether to join or to balk, which is more sensible than the classical viewpoint in queueing theory. Equilibrium customer behavior is considered under four cases: fully observable, almost observable, almost unob- servable and fully unobservable, which cover all the levels of information. Based on the reward-cost structure, we obtain the equilibrium balking strategies in all cases. Further- more, the stationary system behavior is analyzed and a variety of performance measures are developed under the corresponding strategies. Finally, we present several numerical experiments that demonstrate the effect of the information level as well as several param- eters on the equilibrium behavior and social benefit. The research results not only offer the customers optimal strategies but also provide the managers with a good reference to dis- cuss the pricing problem in the queueing system.

Ó2012 Elsevier Inc. All rights reserved.

1. Introduction

Due to the widely applications for management in service system and electronic commerce, there exists an emerging ten- dency to study customers’ behavior in queueing models. In these models, customers are allowed to make decisions as to whether to join or to balk, buy priority or not etc., which is more sensible to describe queueing models. Traditionally, queue- ing systems are divided into the observable model and the unobservable model regarding whether the information of the queue length is available to customers or not prior to their actions. The observable queueing system was first analyzed by Naor[1], who studied equilibrium and social optimal strategies in anM=M=1 queue with a simple linear reward-cost structure. Afterward, Naor’s model and results had been extended in several literatures, see e.g.[2–4]. Chen and Frank[5]

generalized Naor’s model assuming that both the customers and the server maximize their expected discounted utility using a common discount rate. Erlichman and Hassin[6]discussed a priority queue in which customers have the option of over- taking some or all of the customers. On the other hand, Edelson and Hildebrand[7]presented the pioneering literature on the unobservable queue in which the properties of the basic unobservableM=M=1 queue were discovered. Littlechild[8]

extended the model of Edelson and Hildebrand assuming that customers have different service values. Chen and Frank [9]discussed the robustness of the main result of Edelson and Hildebrand, that a profit maximizer chooses a socially optimal admission fee, when the assumption of a linear utility function is removed. Balachandran[10]considered an unobservable

0307-904X/$ - see front matterÓ2012 Elsevier Inc. All rights reserved.

http://dx.doi.org/10.1016/j.apm.2012.08.017

⇑Corresponding author. Tel.: +86 13485330831.

E-mail addresses:myhicharge@126.com(Y. Ma),liuwq@sxu.edu.cn(W.-q. Liu),lijh1982@sxu.edu.cn(J.-h. Li).

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Applied Mathematical Modelling

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / a p m

M=G=1 model with a fixed cost of running the service facility. Subsequently, several authors had investigated the equilibrium strategies in various unobservable models incorporating many diverse characteristics. The fundamental results on this sub- ject in both the observable and unobservable queueing systems can be found in the comprehensive monograph of Hassin and Haviv[11].

Discrete-time queueing systems with vacations have been widely studied in the past because of their extensively use in digital communication and telecommunication networks. An excellent and complete study on discrete-time vacation models had been presented by Takagi[12]. Zhang and Tian[13]presented the detailed analysis on theGeo=G=1 queue with multiple adaptive vacations and further in[14]Tian and Zhang dealt with aGI=Geo=1 queue with multiple vacations and exhaustive service. Recently, Samanta et al.[15]and Tang et al.[16]did research on the discrete-timeGeo^X=G=1 vacation queue with different characteristics.

As for the research on the equilibrium customer behavior in vacation queue models, the first was presented by Burnetas and Economou[17], who explored both the observable and unobservable cases in a single server Markovian queue with set up times. Then, Economou and Kanta[18]analyzed the equilibrium balking strategies in the observable single-server queue with breakdowns and repairs. Recently, Sun et al.[19]considered the equilibrium behavior of customers in an observable M=M=1 queue under interruptible and insusceptible setup/closedown policies. Economou et al.[20]extended the analysis done for the almost and fully unobservable queues in[17]to the non-Markovian case. Liu et al.[21]studied the equilibrium threshold strategies in observable queues under single vacation policy. However, there was no work concerning the equilib- rium balking behavior in the discrete-time queues with multiple vacations.

In the present paper we analyze the equilibrium balking strategies in the discrete-timeGeo=Geo=1 queue with multiple vacations. To the authors’ knowledge, this is the first time that the multiple vacation policy is introduced into the economics of queues. The customers’ dilemma is whether to join the system or balk. They make decisions based on a nature reward-cost structure, which incorporates their desire for service as well as their unwillingness to wait. We explore various cases with regard to the level of information available to customers upon arrival. More specifically, at his arrival epoch a customer may or may not know the number of customers present and/or the state of the server. Therefore, four combinations emerge, rang- ing from full to no information. In each of the four cases we discuss customer equilibrium strategies, analyze the stationary behavior of the corresponding system and derive the equilibrium social benefit for all customers. Furthermore, we present several numerical experiments to explore the effect of the information level as well as several parameters on the equilibrium behavior and the social benefit.

This paper is organized as follows. In Section2, we give the model description and the reward-cost structure. Section3 discusses the observable queue in which customers observe the length of the queue. We distinguish two subcases depending on the additional information, or lack thereof, of the server state. We determine equilibrium threshold strategies and analyze the resulting stationary system behavior. Then Section4studies the unobservable queue where the queue length is not avail- able to customers. We derive the corresponding mixed equilibrium strategies and investigate the stationary behavior for the almost and fully unobservable models. In Section5, we illustrate the effect of the information level on the equilibrium behavior and the social benefit via analytical and numerical comparisons. Finally, in Section 6, we give a necessary conclusion.

2. Model description

We consider a single server queueing system with multiple vacations. The server takes a vacation immediately at the end of each busy period. If he finds the system still empty upon returning from the vacation, he will take another vacation and so on. In this paper, for any real numberx2 ½0;1, we denotex¼1x.

Assume that customer arrivals occur at the end of slott¼n;n¼0;1;. . .. The inter-arrival times are independent and identically distributed sequences following a geometric distribution with ratep.

PðT¼kÞ ¼pp^k1; kP1; 0<p<1; p¼1p:

The service starting and service ending occur at slot division pointt¼n; n¼0;1;. . .. The service times are independent each other and geometrically distributed with rate

l

.

PðS¼kÞ ¼

ll

^k1; kP1; 0<

l

<1;

l

¼1

l

:

The beginning and ending of vacation occur at the epoch which is similar tot¼nin shape. The vacation time is an inde- pendent and identically distributed random variable following a geometric distribution with rateh.

PðV¼kÞ ¼hh^k1; kP1; 0<h<1; h¼1h:

We assume that inter-arrival times, service times, and vacation times are mutually independent. The queueing system fol- lows the First-Come-First-Served (FCFS) service discipline. Moreover, suppose the systems considered are stable. The service rate exceeds the arrival rate so that the server can accommodate all arrivals.

LetLnbe the number of customers in the system at timen^þ. According to the assumptions above, a customer who finishes service and leaves att¼n^þdoes not reckon onLnwhile arrives att¼nshould reckon onLn. We assume

Jn¼ 0; the system is in a vacation period at timen^þ; 1; the system is in a service period at timen^þ:

It is clear thatfLn;J_ngis a Markov chain with state space X¼ fðk;jÞjkPj;j¼0;1g;

where stateðk;0Þ;kP0, indicates that the system is in the vacation period and there arekcustomers; stateðk;1Þ;kP1, indicates that the system is in the busy period and there arekcustomers. Furthermore, we distinguish four cases depending on the information provided to customers before making decisions, which cover all the levels of information.

Fully observable case: Customers observeðLn;J_nÞ;

Almost observable case: Customers observe onlyLn; Almost unobservable case: Customers observe onlyJ_n;

Fully unobservable case: Customers do not observe the system state.

Our interest is in the behavior of customers when they decide whether to join or to balk at their arrival instant. SupposeSe be the mean sojourn time of a customer in equilibrium andBebe the expected net benefit. To model the decision process, we assume that every customer receives a reward ofRunits for completing service. This may reflect his satisfaction and the added value of being served. On the other hand, there exists a waiting cost ofCunits per time unit that the customer remains in the system (in queue or in service). Customers are risk neutral and maximize their expected net benefit. From now on, we assume the condition

R>C

l

^þ

C

h: ð1Þ

This condition ensures that the reward for service exceeds the expected cost for a customer who finds the system empty.

Otherwise, after the system becomes empty for the first time no customers will ever enter. Finally, we stress that the deci- sions are irrevocable: retrials of balking customers and reneging of entering customers are not allowed.

3. Analysis of the observable queues

We first consider the observable queues in which customers are informed of the queue length upon arrival. We show that there exist equilibrium balking strategies of thresholds type. In the fully observable queue, a pure threshold strategy is spec- ified by a pairðL_eð0Þ;L_eð1ÞÞand has the form ‘observeðL_n;J_nÞat arrival instant; enter ifL_n6L_eðJ_nÞand balk otherwise’. In the almost observable queue, a pure threshold strategy is specified by a single numberLeand has the form ’observeLn; enter if Ln6Leand balk otherwise’.

3.1. Fully observable queue

We begin with the fully observable case in which arriving customers know both the number of present customersLnand the state of the serverJ_n. In equilibrium, a customer who joins the system when he observes stateðk;jÞhas mean sojourn time

Se¼kþ1

l

^þ

1j h : Thus his expected net benefit is

Be¼RCðkþ1Þ

l

Cð1jÞ h :

The customer strictly prefers to enter ifBeis positive and is indifferent between entering and balking if it equals zero. We thus conclude the following theorem.

Theorem 3.1. In the fully observable Geo=Geo=1queue with multiple vacations, there exist thresholds ðLeð0Þ;Leð1ÞÞ ¼ R

l

C

l

h

1; R

l

C

1

ð2Þ such that the strategy ‘observeðLn;J_nÞ, enter if Ln6LeðJ_nÞand balk otherwise’ is a unique equilibrium in the class of the threshold strategies.

Remark 1. L_eð0Þis the threshold when an arriving customer finds the system is in a vacation period andL_eð1Þis the thresh- old when it is in a regular busy period. We getLeð0ÞandLeð1Þfrom the conditionBe>0 wheni¼0 and 1 respectively. The symbolbcindicates rounding down.

For the stationary analysis of the system, note that if all customers follow the threshold strategy in (2), the system follows a Markov chain with state space restricted toXfo¼ fðk;0Þj06k6Leð0Þ þ1g [ fðk;1Þj16k6Leð1Þ þ1g. The transition rate diagram is depicted inFig. 1. The one-step transition probabilities ofðLn;J_nÞare as follows:

Case 1: ifXn¼ ð0;0Þ,

Xnþ1¼

ð0;0Þ; with probabilityp;

ð1;0Þ; with probabilityhp;

ð1;1Þ; with probabilityhp:

8>

<

>:

Case 2: ifXn¼ ðk;0Þ;16k6Leð0Þ,

Xnþ1¼

ðk;0Þ; with probabilityhp;

ðk;1Þ; with probabilityhp;

ðkþ1;0Þ; with probabilityhp;

ðkþ1;1Þ; with probabilityhp:

8>

>>

<

>>

>:

Case 3: ifXn¼ ðLeð0Þ þ1;0Þ,

Xnþ1¼ ðLeð0Þ þ1;0Þ; with probabilityh;

ðLeð0Þ þ1;1Þ; with probabilityh:

(

Case 4: ifXn¼ ð1;1Þ,

Xnþ1¼

ð0;0Þ; with probabilityp

l

;

ð1;1Þ; with probability 1p

l

p

l

; ð2;1Þ; with probabilityp

l

:

8>

<

>:

Case 5: ifXn¼ ðk;1Þ;26k6Leð1Þ

Xnþ1¼

ðk1;1Þ; with probabilityp

l

;

ðk;1Þ; with probability 1p

l

p

l

; ðkþ1;1Þ; with probabilityp

l

:

8>

<

>:

Case 6: ifXn¼ ðLeð1Þ þ1;1Þ,

Xnþ1¼ ðLeð1Þ;1Þ; with probability

l

; ðLeð1Þ þ1;1Þ; with probability

l

:

Based on the one-step transition situation analysis, using the lexicographical sequence for the states, the transition prob- ability matrix can be written as

...

...

...

0 ), 0

e(

L Le(0)1,0 1 , 1 ) 0

e( L 1 ), 0

e(

L L_e(0) 2,1 L_e(1),1 L_e(1) 1,1

p p

p p

p

p p p 1

p

p ^p p p ^p p

p p p p p p p

p p p p p p p

p p

p p p p

p p

1 1 p p 1 p p 1 p p 1 p p

Fig. 1.Transition rate diagram for theðLeð0Þ;Leð1ÞÞthreshold strategy in the fully observable queue.

eP¼

B0 A0

B1 A1 C1

B2 A1 C1

.. . .. .

.. . B2 A1 C1

B2 A2 C2

B3 A3 C3

B4 A3 C3

.. . .. .

.. . B4 A3 C3

B5 A4

2 66 66 66 66 66 66 66 66 66 66 66 64

3 77 77 77 77 77 77 77 77 77 77 77 75

; ð3Þ

where

B0¼p; A0¼ ðhp;hpÞ;

B1¼ 0 p

l

; A1¼ hp hp

0 1p

l

p

l

" #

; C1¼ hp hp

0 p

l

" #

;

B2¼ 0 0 0 p

l

; A2¼ h h

0 1p

l

p

l

" #

; C2¼ 0

p

l

and

B3¼ ð0;p

l

Þ; A3¼1p

l

p

l

; C3¼p

l

; B4¼p

l

; A4¼

l

; B5¼

l

: LetðL;JÞbe the stationary limit ofðLn;J_nÞand its distribution is denoted as

p

kj¼PfL¼k;J¼jg; ðk;jÞ 2Xfo

p

k¼

p

00; k¼0;

ð

p

k0;

p

k1Þ; 16k6Leð0Þ þ1;

p

k1; Leð0Þ þ26k6Leð1Þ þ1:

8>

<

>:

p

¼ ð

p

0;

p

1;. . .;

p

Leð1Þþ1Þ:

We solve for the stationary distribution

p

kjby noting that the vector

p

satisfies the equation

p

^eP¼

p

and have the following system of steady-state equations:

p

00¼p

p

00þp

lp

11; ð4Þ

p

k0¼hp

p

k1;0þhp

p

k0; k¼1;2;3;. . .;Leð0Þ; ð5Þ

p

Leð0Þþ1;0¼hp

p

Leð0Þ;0þh

p

Leð0Þþ1;0; ð6Þ

p

11¼hp

p

00þhp

p

10þ ð1p

l

p

l

Þ

p

11þp

lp

21; ð7Þ

p

k1¼hp

p

k1;0þp

lp

k1;1þhp

p

k0þ ð1p

l

p

l

Þ

p

k1þp

lp

kþ1;1; k¼2;3;. . .;Leð0Þ; ð8Þ

p

Leð0Þþ1;1¼hp

p

Leð0Þ;0þp

lp

Leð0Þ;1þh

p

Leð0Þþ1;0þ ð1p

l

p

l

Þ

p

Leð0Þþ1;1þp

lp

Leð0Þþ2;1; ð9Þ

p

k1¼p

lp

k1;1þ ð1p

l

p

l

Þ

p

k1þp

lp

kþ1;1; k¼Leð0Þ þ2;. . .;Leð1Þ 1; ð10Þ

p

Leð1Þ;1¼p

lp

Leð1Þ1;1þ ð1p

l

p

l

Þ

p

Leð1Þ;1þ

lp

Leð1Þþ1;1; ð11Þ

p

Leð1Þþ1;1¼p

lp

Leð1Þ;1þ

lp

Leð1Þþ1;1: ð12Þ

Define

a

¼^p_p^l_l;b¼ ^hp

1hp. Thus

a

<1 andb<1.

By iterating(5) and (10), taking into account(6), (11) and (12), we obtain

p

k0¼b^k

p

00; k¼1;2;. . .;Leð0Þ; ð13Þ

p

Leð0Þþ1;0¼1hp

h b^Leð0Þþ1

p

00; ð14Þ

p

k1¼

a

^kLeð0Þ1

p

Leð0Þþ1;1; k¼Leð0Þ þ2;. . .;Leð1Þ; ð15Þ

p

Leð1Þþ1;1¼p

a

^{Leð1ÞLeð0Þ}

p

Leð0Þþ1;1: ð16Þ

From(8)we observe thatf

p

k1jk¼1;2;. . .;Leð0Þ þ1gis a solution of the nonhomogeneous linear difference equation with constant coefficients

p

l

xkþ1 ðp

l

þp

l

Þxkþp

l

xk1¼ hp

p

k1;0hp

p

k0¼ h

hb^k

p

00; k¼2;3;. . .;Leð0Þ; ð17Þ

where the last equation is due to(13). Using the standard approach for solving such equations, see e.g.[22], we consider the corresponding characteristic equation

p

l

x² ðp

l

þp

l

Þxþp

l

¼0;

which has two roots at 1 and

a

. Then the general solution of the homogeneous version of(17)isx^hom_k ¼A1^kþB

a

^k. The general solutionx^gen_k of(17)is given asx^gen_k ¼x^hom_k þx^spec_k , wherex^spec_k is a specific solution of(17). Because the nonho- mogeneous part of(17)is geometric with parameterb, we can find a specific solutionDb^k(we assume

a

–b). Substituting x^gen_k ¼Db^kinto(17)we obtain

D¼1hp

hp

l p

00: ð18Þ

Hence the general solution of(17)is given as

x^gen_k ¼A1^kþB

a

^kþDb^k; k¼1;2;. . .;Leð0Þ þ1; ð19Þ

whereDis given by(18)andA;Bare to be determined.

From(19)fork¼1, we obtain AþB

a

þDb¼

p

11¼ p

p

l p

00: ð20Þ

Furthermore, substituting(19)into(7)fork¼2, it follows after some rather tedious algebra that AþB

a

²þDb²¼

p

21¼p²ðp

l

þp

l

hþph

l

Þ

ðp

l

Þ²ð1hpÞ

p

00: ð21Þ

Solving the system of(20) and (21), we obtain A¼0; B¼ hhp

l

hhp

p

00: Then, from(19),

p

k1¼ hþhp

l

hhpðb^k

a

^kÞ

p

00; k¼1;2;. . .;Leð0Þ þ1: ð22Þ

We have thus expressed all stationary probabilities in terms of

p

00in relations see(13)–(16)and(22). The remaining prob- ability

p

00can be found from the normalization equation

X

Leð0Þþ1

k¼0

p

k0þ^LeX^ð1Þþ1

k¼1

p

k1¼1:

After some algebraic simplification, we can express all stationary probabilities in the following theorem.

Theorem 3.2.Consider a fully observable Geo/Geo/1 queue with multiple vacations and

a

–b, in which customers follow the threshold policyðLeð0Þ;Leð1ÞÞgiven inTheorem3.1. The stationary probabilitiesf

p

kjjðk;jÞ 2Xfogare as follows:

p

00¼ hþhp

h þ hþhp

l

hhp hp

h p

l l

pþ pp

l

p

a

^Leð1Þþ1þ p

l l

php

h pp

l

p

a

^{Leð1ÞLeð0Þ}

b^Leð0Þþ1

¹

; ð23Þ

p

k0¼b^k

p

00; k¼1;2;. . .;Leð0Þ; ð24Þ

p

Leð0Þþ1;0¼b^Leð0Þþ1

1b

p

00; ð25Þ

p

k1¼ hþhp

l

hhpðb^k

a

^kÞ

p

00; k¼1;2;. . .;Leð0Þ þ1; ð26Þ

p

k1¼ hþhp

l

hhp b

a

Leð0Þþ1

1

" #

a

^k

p

00; k¼Leð0Þ þ2;. . .;Leð1Þ; ð27Þ

p

Leð1Þþ1;1¼ hþhp

l

hhp b

a

Leð0Þþ1

1

" #

p

a

^Leð1Þþ1

p

00: ð28Þ

Because the probability of balking is equal to

p

Leð0Þþ1;0þ

p

Leð1Þþ1;1, the social benefit per time unit when all customers fol- low the threshold policyðLeð0Þ;Leð1ÞÞgiven inTheorem 3.1equals

SBfo¼Rpð1

p

Leð0Þþ1;0

p

Leð1Þþ1;1Þ C ^LeX^ð0Þþ1

k¼0

k

p

k0þ^LeX^ð1Þþ1

k¼1

k

p

k1

! :

3.2. Almost observable queue

We next consider the almost observable case, where arriving customers only know the queue lengthLnbefore making decisions. Hence the stationary distribution of the corresponding Markov chain is from Theorem 3.2 with Leð0Þ ¼Leð1Þ ¼Leand state spaceXao¼ fkj06k6Leþ1g. The transition rate diagram is depicted inFig. 2.

Theorem 3.3. Consider an almost observable Geo/Geo/1 queue with multiple vacations and

a

–b, in which customers follow the threshold policy Le. The stationary probabilitiesf

p

⁰_kjk2Xaogare as follows:

p

⁰₀¼ hþhp

l

hhp

l

²

l

p

l

hp hð

l

pÞ þ p

l

l

p

a

^Leþ1

l

hb^Leþ1

¹

;

p

⁰_k¼

l

l

hhpb^k hþhp

l

hhp

a

^k

p

⁰₀; k¼1;2;. . .;Le;

p

⁰_Leþ1¼ ð

l

hÞðhþhpÞ

hð

l

hhpÞ b^Leþ1 pðhþhpÞ

l

hhp

a

^Leþ1

" #

p

⁰₀:

Because the expected net benefit of a customer who findskcustomers in the system, if he decides to enter, is

Be¼RCðkþ1Þ

l

C

p

_JjLð0jkÞ

h ; ð29Þ

where

p

_JjLð0jkÞis the probability that an arriving customer finds the server in a vacation time, given that there arekcustom- ers. Using the various forms of

p

kjfrom(23)–(28), we get

p

_JjLð0jkÞ ¼ 1þ_lhhp^hþhp 1 ^a_b ^k

1

; k¼0;1;2;. . .;Le;

p

_JjLð0jLeþ1Þ ¼ 1þ ð1bÞp_l^hþhp_hhp 1 ^a_b ^Leþ1

1

: 8>

>>

<

>>

>:

ð30Þ

In light of(29) and (30), we introduce the function gðk;yÞ ¼RCðkþ1Þ

l

C

h 1þy hþhp

l

hhp 1

a

b

k!

" #1

; y2 ½ð1bÞp;1; k¼0;1;2;. . .; ð31Þ which will allow us to prove the existence of equilibrium threshold strategies and derive the corresponding thresholds. Let

g_UðkÞ ¼gðk;1Þ ¼RCðkþ1Þ

l

C

h 1þ hþhp

l

hhp 1

a

b

k!

" #1

; k¼0;1;2;. . .; ð32Þ

gLðkÞ ¼gðk;ð1bÞpÞ ¼RCðkþ1Þ

l

C

h 1þ ð1bÞp hþhp

l

hhp 1

a

b

k!

" #1

; k¼0;1;2;. . .; ð33Þ

...

...

L_e,0

1

e ,

L L_e 1,1

0 ,

e 1 L p

p

1 1 p p 1 p p

p p p p

p p p

p

p p p

p p p p p

p p

p

p p p p p

p

Fig. 2.Transition rate diagram for theLethreshold strategy in the almost observable queue.

It is easy to see that g_Uð0Þ ¼g_Lð0Þ ¼RC

l

C h>0:

In addition,

limk!1g_UðkÞ ¼lim

k!1g_LðkÞ ¼ 1:

Hence there existskU such that

gUð0Þ;gUð1Þ;gUð2Þ;. . .;gUðkUÞ>0 andgUðkUþ1Þ60: ð34Þ

Because the functiongðk;yÞis increasing with respect toyfor every fixedk, we get the relationg_LðkÞ6g_UðkÞ; k¼0;1;2;. . ..

In particular,g_LðkUþ1Þ60 whileg_Lð0Þ>0. Hence, there existskL6kUsuch that

g_LðkLÞ>0 andg_LðkLþ1Þ;. . .;g_LðkUÞ;g_LðkUþ1Þ60: ð35Þ We can now establish the existence of the equilibrium threshold policies in the almost observable case and give the follow- ing theorem.

Theorem 3.4.In the almost observable Geo=Geo=1queue with multiple vacations, all pure threshold strategies ‘observe Ln, enter if Ln6Leand balk otherwise’ for Le¼kL;kLþ1;. . .;kUare equilibrium balking strategies.

Proof.Consider a tagged customer at his arrival instant and assume all other customers follow the same threshold strategy

‘observeLn, enter ifLn6Leand balk otherwise’ for some fixedLe2 fkL;kLþ1;. . .;kUg. Then

p

_JjLð0jkÞis given by(30).

If the tagged customer findsk6Lecustomers and decides to enter, his expected net benefit is equal to RCðkþ1Þ

l

C

h 1þ hþhp

l

hhp 1

a

b

k!

" #1

¼g_UðkÞ>0;

because of(29)–(32)and(34). So in this case the customer prefers to enter.

If the tagged customer findsk¼Leþ1 customers and decides to enter, his expected net benefit is RCðLeþ2Þ

l

C

h 1þ ð1bÞp hþhp

l

hhp 1

a

b

Leþ1!

" #1

¼g_LðLeþ1Þ60;

because of(29)–(31),(33) and (35). Therefore in this case the customer prefers to balk. h

Remark: There exist equilibrium mixed threshold strategies. InTheorem 3.4, we restrict our attention to the pure thresh- old strategies because they are evolutionarily stable strategy (ESS) while the mixed are not ESS.

Because the probability of balking is equal to

p

⁰_Leþ1, the social benefit per time unit when all customers follow the thresh- old policyLegiven inTheorem 3.4equals

SBao¼Rpð1

p

⁰_Leþ1Þ C X^Leþ1

k¼0

k

p

⁰_k

! :

4. Analysis of the unobservable queues

In this section we turn attention to the unobservable queues, where arriving customers do not observe the length of the queue. We prove that there exist equilibrium mixed strategies. In the almost unobservable queue, a mixed strategy is spec- ified by a vectorðqð0Þ;qð1ÞÞ, whereqðjÞis the probability of joining when the server is in statej. In the fully unobservable queue, where customers are provided with no information, a mixed strategy is specified by the probabilityqof entering.

4.1. Almost unobservable queue

We begin with the almost unobservable case in which arriving customers observe the statejof the server upon arrival. If all customers follow the same mixed strategyðqð0Þ;qð1ÞÞ, then the system follows a Markov chain in which the arrival rate equalspðjÞ ¼pqðjÞwhen the server is in statej. The state space isXau¼ fðk;jÞjkPj;j¼0;1gand the transition rate diagram is illustrated inFig. 3.

Using the lexicographical sequence for the states, the transition probability matrix can be written as

eP¼

B0 A0

B1 A1 C1

B2 A1 C1

B2 A1 C1

.. . .. .

.. . 2

66 66 66 64

3 77 77 77 75

; ð36Þ

where

B0¼pð0Þ; A0¼ ðhpð0Þ;hpð0ÞÞ; B1¼ 0 pð1Þ

l

;

B2¼ 0 0 0 pð1Þ

l

; A1¼ hpð0Þ hpð0Þ 0 1pð1Þ

l

pð1Þ

l

" #

; C1¼ hpð0Þ hpð0Þ 0 pð1Þ

l

" #

:

Due to the block tridiagonal structure of transition probability matrix,fLn;J_ngis a quasi birth and death chain. LetðL;JÞbe the stationary limit ofðLn;J_nÞand its distribution is denoted as

p

⁰⁰_kj¼PfL¼k;J¼jg; ðk;jÞ 2Xau;

p

⁰⁰₀¼

p

⁰⁰₀₀;

p

⁰⁰_k¼ ð

p

⁰⁰_k0;

p

⁰⁰_k1Þ; kP1;

p

⁰⁰¼ ð

p

⁰⁰₀;

p

⁰⁰₁;

p

⁰⁰₂;. . .Þ:

We solve forð

p

⁰⁰₁₀;

p

⁰⁰₁₁Þby the equation

p

⁰⁰Pe¼

p

⁰⁰and obtain that:

ð

p

⁰⁰₁₀;

p

⁰⁰₁₁Þ ¼ hpð0Þ 1hpð0Þ; pð0Þ

pð1Þ

l

!

p

⁰⁰₀₀: ð37Þ

Using the matrix-geometric solution method, the rate matricRis the solution of the matrix quadratic equation R¼R²B2þRA1þC1:

After some rather tedious algebra, we obtain that:

R¼ bð0Þ ^pð0Þ

pð1Þl 0

a

ð1Þ

" #

; ð38Þ

wherebð0Þ ¼ ^hpð0Þ

1hpð0Þ;

a

ð1Þ ¼^pð1Þl

pð1Þl. Thus from

p

⁰⁰_k¼ ð

p

⁰⁰₁₀;

p

⁰⁰₁₁ÞR^k1; kP1; ð39Þ

we can express all stationary probabilities in terms of

p

⁰⁰₀₀in relations see(37)–(39). The remaining probability

p

⁰⁰₀₀can be found from the normalization equation

X¹

k¼0

p

⁰⁰_k0þX¹

k¼1

p

⁰⁰_k1¼1:

After some algebraic simplification, we can express all stationary probabilities in the following theorem.

...

...

1 ,

k

k 1,1

0 , 1 k )

1 ( ) 1 (

1 p p

) 1 ( p

) 1 ( p

) 0 ( ) p 0 ( p

) 0 ( p ) 0 ( p ) 0 ( p

) 0 (

p p(0) p(0)

k , 0

p(0)

) 0 (

p _p(0) p(0) p(0) p(0) p(0) p(0) )

1 (

p p(1) p(1) p(1)

) 1 (

p p(1) p(1)

) 1 ( ) 1 (

1 p p 1 p(1) p(1) 1 p(1) p(1)

) 1 ( p

) 1 ( p

) 0 ( p ) 0 ( p

...

...

) 0 (

p p(0) p(0)

Fig. 3.Transition rate diagram for theðqð0Þ;qð1ÞÞmixed strategy in the almost unobservable queue.

Theorem 4.1. Consider a Geo/Geo/1 queue with multiple vacations andbð0Þ–

a

ð1Þ, in which customers observe the state j of the server upon arrival and enter with probability qðjÞ, i.e. they follow the mixed policyðqð0Þ;qð1ÞÞ. The stationary probabilities f

p

⁰⁰_kjjðk;jÞ 2Xaugare

p

⁰⁰₀₀¼ hð

l

pð1ÞÞ

ðhþhpð0ÞÞð

l

pð1Þ þpð0ÞÞ; ð40Þ

ð

p

⁰⁰₁₀;

p

⁰⁰₁₁Þ ¼ hpð0Þ 1hpð0Þ; pð0Þ

pð1Þ

l

!

p

⁰⁰₀₀; ð41Þ

ð

p

⁰⁰_k0;

p

⁰⁰_k1Þ ¼ ð

p

⁰⁰₁₀;

p

⁰⁰₁₁ÞR^k1; kP1: ð42Þ FromTheorem 4.1, we can easily obtain the state probabilities of a server in steady-state

PðJ¼0Þ ¼X¹

k¼0

p

⁰⁰_k0¼

l

pð1Þ

l

pð1Þ þpð0Þ; ð43Þ

PðJ¼1Þ ¼X¹

k¼1

p

⁰⁰_k1¼ pð0Þ

l

pð1Þ þpð0Þ: ð44Þ

Now we consider a customer who finds the server at statejupon arrival, thus his mean sojourn time is SeðjÞ ¼E½Ljj þ1

l

^þ

1j h ;

whereE½Ljjis the expected number of customers in system found by an arrival, given that the server is found at statej. The expected net benefit of such a customer who decides to enter is

BeðjÞ ¼RCðE½Ljj þ1Þ

l

Cð1jÞ

h : ð45Þ

We thus need to computeE½Ljjwhen all customers follow the same mixed strategy ðqð0Þ;qð1ÞÞ. Since the probability

p

_LjJðkjjÞ, that an arrival findskcustomers in system, given that the server is found at statejis

p

_Lj0ðkj0Þ ¼_PðJ¼0Þ^p00k0 ; k¼0;1;2;. . .;

p

_Lj1ðkj1Þ ¼_PðJ¼1Þ^p00k1 ; k¼1;2;. . .: 8<

: ð46Þ

Substituting(40)–(44)into(46)and fromE½Ljj ¼P1

k¼jk

p

_LjJðkjjÞ, we get E½Lj0 ¼hpð0Þ

h ; ð47Þ

E½Lj1 ¼hpð0Þ

h þ

l

pð1Þ

l

pð1Þ: ð48Þ

Then the social benefit per time unit when all customers follow the mixed policyðqð0Þ;qð1ÞÞcan be easily computed as

SBau¼p

l

pð1Þ

l

pð1Þ þpð0Þqð0Þ RC

l

hþhpð0Þ

h C

h

þp pð0Þ

l

pð1Þ þpð0Þqð1Þ RC

l

hþhpð0Þ

h þ

l

pð1Þ

l

pð1Þ

!

" #

:

Substituting(47) and (48)into(45)we can identify mixed equilibrium strategies for the almost unobservable model.

Theorem 4.2. In the almost unobservable Geo=Geo=1 queue with multiple vacations, there exists a unique mixed strategy ðq_eð0Þ;q_eð1ÞÞ‘observe J_nand enter with probability q_eðJ_nÞ’ where the vectorðq_eð0Þ;q_eð1ÞÞis given as follows:

Case I: ¹_h<_l¹,

ðqeð0Þ;qeð1ÞÞ ¼

1 hp

lhR

C

l

h

;0

; R2^C_lþ^C_h;^CðhþhpÞ_lh þ^C_h

; ð1;0Þ; R2h^CðhþhpÞ_lh þ^C_h;^C_lþ^CðhþhpÞ_lh

; 1;pðCpþChCphþC^{lðCpþ2ChCph}lh^lhRÞlhRÞ

; R2h^C_lþ^CðhþhpÞ_lh ;_l^Cp_pþ^CðhþhpÞ_lh

; ð1;1Þ; R2h_l^Cp_pþ^CðhþhpÞ_lh ;1

: 8>

>>

>>

>>

>>

><

>>

>>

>>

>>

>>

:

Case II: ¹_l6¹

h6 ^p lp,

ðqeð0Þ;qeð1ÞÞ ¼

1 hp

lhR

C

l

h

;^lh

hp

; R2 _l^Cþ^C_h;^CðhþhpÞ_l

h þ^C_h

; 1;pðCpþChCphþC^{lðCpþ2ChCph}lh^lhRÞlhRÞ

; R2h^CðhþhpÞ_lh þ^C_h;_l^Cp_pþ^CðhþhpÞ_lh

; ð1;1Þ; R2h_l^Cp_pþ^CðhþhpÞ_lh ;1

: 8>

>>

><

>>

>>

: Case III: _lp^p <¹_h,

ðq_eð0Þ;q_eð1ÞÞ ¼

1 hp

lhR

C

l

h

;1

; R2^C_lþ^C_h;^CðhþhpÞ_lh þ^C_h

;

ð1;1Þ R2h^CðhþhpÞ_lh þ^C_h;1

: 8>

<

>:

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

Beð0Þ ¼RCðE½Lj0 þ1Þ

l

C

h¼RCðhþhpð0ÞÞ

l

h C h: Therefore we have two cases:

Case 1:Beð0Þ60i:e:_l^Cþ^C_h<R6^CðhþhpÞ lh þ^C_h.

In this case if all customers who find the system empty enter with probabilityq_eð0Þ ¼1, then the tagged customer suffers a negative expected benefit if he decides to enter. Henceq_eð0Þ ¼1 does not lead to an equilibrium. Similarly, if all customers useq_eð0Þ ¼0, then the tagged customer receives a positive benefit from entering. Thusq_eð0Þ ¼0 also cannot be part of an equilibrium mixed strategy. Therefore, there exists a uniqueq_eð0Þsatisfying

RCðhþhpqeð0ÞÞ

l

h C h¼0;

for which customers are indifferent between entering and balking. This is given by q_eð0Þ ¼ 1

hp

l

hR C

l

h

: ð49Þ

Case 2:Beð0Þ>0i:e:R>^CðhþhpÞ_lh þ^C_h.

In this case, for every strategy of the other customers, the tagged customer has a positive expected net benefit if he decides to enter. Thereforeqeð0Þ ¼1.

We next considerq_eð1Þand tag a customer who finds the server at state 1 upon arrival. If he decides to enter, his expected net benefit is

Beð1Þ ¼RCðE½Lj1 þ1Þ

l

^¼R

Cpð1Þ

l

pð1ÞCðhþhpð0ÞÞ

l

h ¼

C

h_l^Cpð1Þ_pð1Þ; in case 1;

R_l^Cpð1Þ_pð1Þ^CðhþhpÞ_lh ; in case 2:

8<

: ð50Þ

Therefore to findq_eð1Þin equilibrium, we must examine Case 1 and Case 2 separately and consider the following subcases in each:

Case 1a: ^C_lþ^C_h<R6^CðhþhpÞ

lh þ^C_h and ^C_h<^C_l. ðq_eð0Þ;q_eð1ÞÞ ¼ 1

hp

l

hR C

l

h

;0

: Case 1b: ^C_lþ^C_h<R6^CðhþhpÞ

lh þ^C_h and _l^C6^C

h6 ^Cp lp. ðqeð0Þ;qeð1ÞÞ ¼ 1

hp

l

hR C

l

h

;

l

h

hp

: Case 1c: ^C_lþ^C_h<R6^CðhþhpÞ

lh þ^C_h and _l^Cp_p<^C_h. ðq_eð0Þ;q_eð1ÞÞ ¼ 1

hp

l

hR C

l

h

;1

: Case 2a: R>^CðhþhpÞ_lh þ^C_h and R<^C_lþ^CðhþhpÞ_lh .

ðq_eð0Þ;q_eð1ÞÞ ¼ ð1;0Þ:

Case 2b: R>^CðhþhpÞ_lh þ^C_h and ^C_lþ^CðhþhpÞ_lh 6R6 ^Cp

lpþ^CðhþhpÞ_lh . ðq_eð0Þ;q_eð1ÞÞ ¼ 1;

l

ðCpþ2ChCph

l

hRÞ

pðCpþChCphþC

l

h

l

hRÞ

:

Case 2c: R>^CðhþhpÞ_lh þ^C_h and R>_l^Cp_pþ^CðhþhpÞ_lh . ðqeð0Þ;qeð1ÞÞ ¼ ð1;1Þ:

By rearranging Cases 1a–2c asRvaries from_l^Cþ^C_hto infinity, keeping the operating parametersp;

l

;hand the waiting cost rateCfixed, we obtain Case I-III in the theorem statement. h

It seems reasonable at first glance that arriving customers are less willing to enter the system when they find the server on vacation, since they have to wait for the left vacation time, i.e., we might expect thatq_eð0Þ6q_eð1Þ. However, this is not generally true. AsTheorem 4.2shows: in case I it is always true thatq_eð0ÞPq_eð1Þ. In fact, consider a system with small mean vacation time and concentrate on a tagged customer. If he is given the information that the server is on vacation, then he knows that he must wait for the left vacation time. On the other hand, he expects that few customers are ahead of him, be- cause the system is on vacation and the mean vacation time is small. Thus it is optimal for the tagged customer to enter.

4.2. Fully unobservable queue

We finally consider the fully unobservable case, where customers observe neither the state of the system nor the queue length. Here a mixed strategy for a customer is specified by the probabilityqof entering. The stationary distribution of the system state is fromTheorem 4.1by takingqð0Þ ¼qð1Þ ¼qand the state spaceXfuis identical withXau. The transition rate diagram is depicted inFig. 4, wherep⁰¼pq.

Theorem 4.3. Consider a fully unobservable Geo=Geo=1queue with multiple vacations andb–

a

, in which customers enter with probability q, i.e. they follow the mixed policy q. The stationary probabilitiesf

p

⁰⁰⁰_kjjðk;jÞ 2Xfugare

p

⁰⁰⁰₀₀¼ hð

l

p⁰Þ ðhþhp⁰Þ

l

^;

ð

p

⁰⁰⁰₁₀;

p

⁰⁰⁰₁₁Þ ¼ hp⁰ 1hp⁰; p⁰

p⁰

l

p

⁰⁰⁰₀₀; ð

p

⁰⁰⁰_k0;

p

⁰⁰⁰_k1Þ ¼ ð

p

⁰⁰⁰₁₀;

p

⁰⁰⁰₁₁ÞRe^k1; kP1;

where

a

¼p⁰

l

p⁰

l

^{; b¼}

hp⁰

1hp⁰;Re¼ b _p^p₀⁰_l 0

a

" # :

From(43), (44), (47) and (48)withqð0Þ ¼qð1Þ ¼q, we can easily get:

PðJ¼0Þ ¼

l

p⁰

l

^{; PðJ¼1Þ ¼}

p⁰

l

^{; ð51Þ}

E½Lj0 ¼hp⁰

h ; E½Lj1 ¼hp⁰ h þ

l

p⁰

l

p⁰: ð52Þ

...

... k , 0

1 , 1 k

0 , 1 k

'

1 p' p p'

p'

p'

p'

p'

p'

1 ,

k

...

...

p'

p' p^' p^'

p' p^' p^' p^' p^'

p' p^' p^' p^' p^' p^' p^' p^'

p' p^' p^' p^' p^'

p' p^' p^' p^'

'

1 p' p 1 p^' p^' 1 p^' p^'

Fig. 4.Transition rate diagram for theqmixed strategy in the fully unobservable queue.

Hence,

SBfu¼p⁰ RC

l

l

þhhp⁰ h þ p⁰p⁰

l

p⁰

:

Theorem 4.4. In the fully unobservable Geo=Geo=1 queue with multiple vacations, there exists a unique mixed equilibrium strategy ‘enter with probability q_e’, where q_eis given by

q_e¼

lhRClCh

pðRhCChÞ R2^C_hþ^C_l;^C_hþ_l^Cp_p

; 1 R2h^C_hþ_l^Cp_p;1

: 8>

<

>: ð53Þ

Proof. We consider a tagged customer at his arrival instant. If he decides to enter his mean sojourn time Se¼E½L þ1

l

^þ

PðJ¼0Þ

h :

From(51) and (52), we get E½L ¼hp⁰

h þ p⁰p⁰

l

p⁰: Hence the expected net benefit

Be¼RCð1pqÞ

l

pq C

h: ð54Þ

When R2^C_hþ^C_l;^C_hþ_l^Cp_p

, we find (54) has a unique root in ð0;1Þ which gives the first branch of (53). When R2h^C_hþ_l^Cp_p;1

;Beis positive for everyq, thus the best response is 1 and the unique equilibrium point isq_e¼1, which gives the second branch of(53). h

5. Numerical examples

In this section, according to the analysis above, we firstly present a set of numerical experiments to show the effect of the information level as well as several parameters on the behavior of the system. And then we give an example to show the use of results.

5.1. Numerical experiments

Here we concern about the values of the equilibrium thresholds for the observable systems and the values of the equi- librium entrance probabilities for the unobservable systems as well as the social benefit per unit time when customers fol- low the corresponding equilibrium strategies.

We first consider the observable models and explore the sensitivity of the equilibrium thresholds with respect to the ar- rival ratep, vacation ratehand service rewardR. From the three sub-figuresða1;b1;c1Þon the left ofFig. 5, we can make an interesting conjecture that the equilibrium thresholds fkL;. . .;kUg for the almost observable case always locate in ðLeð0Þ;Leð1ÞÞfor the fully observable case. In other words, the thresholds in the almost observable model have intermediate values between the two separate thresholds when arriving customers observe the state of the server.

Concerning the sensitivity of the equilibrium thresholds, we can make the following observations. When the arrival rate pvaries, the fully observable thresholds remain fixed since the arrival rate is irrelevant to the customer’s decision when he has full state information. On the other hand, the almost observable thresholds increase with the arrival rate, which means that if an arriving customer is told the information of the present queue length, then he is more likely to enter when the arrival rate is higher. This phenomenon in the almost observable case that customers in equilibrium tend to imitate the behavior of other customers is of the ‘Follow-The-Crowd’ (FTC) type. The reason for this phenomenon is that when the arrival rate is high, it is probably that the server is active, therefore the expected delay from server vaca- tion is reduced. However, all thresholds increase when the vacation ratehvaries, except thatLeð1Þremains constant. This is certainly intuitive, because when the server vacation is shorter, customers generally have a greater incentive to enter both in the fully and almost observable systems. Finally, along with the increasing of the service rewardR, no matter in the fully observable model or in the almost observable model, the thresholds increase in a linear fashion, which is ex- pected from Theorems 3.1 and 3.4.

0 0.2 0.4 0.6 0.8 1 0

5 10 15 20 25 30

thresholds

p

a

₁

L_e(0) Le(1) k_U k_L

0 0.2 0.4 0.6 0.8 1

0 0.2 0.4 0.6 0.8 1

entrance probabilities

p

a

₂

q_e(0) q_e(1) q_e

0 0.2 0.4 0.6 0.8 1

0 5 10 15 20 25 30

thresholds

θ

b

₁

L_e(0) L_e(1) k_U kL

0 0.1 0.2 0.3 0.4 0.5

0 0.2 0.4 0.6 0.8 1

b

₂

entrance probabilities

θ

q_e(0) q_e(1) q_e

25 30 35 40

0 5 10 15 20 25 30 35

c

₁

thresholds

R

L_e(0) L_e(1) kU k_L

20 25 30 35 40

0 0.2 0.4 0.6 0.8 1

c

₂

entrance praobabilities

R

qe(0) q_e(1) q_e

Fig. 5.Equilibrium thresholds for the observable systems. Sensitivity with respect to: a1. p, for l¼0:95;h¼0:05;C¼1;R¼30;b1. h, for p¼0:4;l¼0:9;C¼1;R¼30;c1.R, forp¼0:4;l¼0:9;h¼0:05;C¼1. Equilibrium entrance probabilities for the unobservable systems. Sensitivity with respect to:a2.p, forl¼0:95;h¼0:3;C¼1;R¼4:5;b2.h, forp¼0:9;l¼0:95;C¼1;R¼8;c2.R, forp¼0:4;l¼0:5;h¼0:05;C¼1.

Next we turn to the unobservable systems and explore the sensitivity of the equilibrium entrance probabilities. The re- sults are presented in the three sub-figuresða2;b2;c2Þon the right ofFig. 5. A general observation is that the entrance prob- ability in the fully unobservable queueing system is always inside the interval formed by the two entrance probabilities in the almost unobservable queueing system, which is similar to the results in the observable models. Therefore when custom- ers are not told the server state, they join the queue with a probability between those in the two separate cases provided that the state of the server can be observed by arriving customers. With regard to the sensitivity of entrance probabilities, we observe that they are decreasing with respect top. Therefore when the arrival rate increases, customers are less willing to enter the system. This is in contrast to the observable models, where thresholds are increasing withp. The reason for the difference is that here the information about the present queue length is not available, when the arrival rate is higher, arriving customers expect that the system is more loaded and are less inclined to enter. Furthermore the entrance probabil- ities are all increasing with respect tohandR, which is intuitive.

Figs. 6–8are concerned with the social benefit under the equilibrium balking strategy for the different information levels.

For the almost observable case, we only present the social benefit under the two extreme thresholdskLandkU. In figures we notice that the social benefit in the almost observable model is generally more than those in other cases while the value for the fully observable model is always in an intermediate position. In addition, the difference in social benefit is small between the fully unobservable and almost unobservable systems. These phenomena indicate that the customers as a whole are gen- erally better off when they are informed of the number of present customers while the information about the server state is

0 0.2 0.4 0.6 0.8 1

0 10 20 30 40 50

60 Equilibrium Social Benefit

p

F.O.

A.O.k_L A.O.k_U A.U.

F.U.

Fig. 6.Social benefit for different information levels. Sensitivity with respect top, forl¼0:99;h¼0:05;R¼80;C¼1.

0 0.1 0.2 0.3 0.4 0.5

0 5 10 15 20 25

30 Equilibrium Social Benefit

θ

F.O.

A.O.k_L A.O.k

U A.U.

F.U.

Fig. 7.Social benefit for different information levels. Sensitivity with respect toh, forl¼0:99;p¼0:2;R¼110;C¼1.

not so beneficial. Moreover, the full information about the queueing system may actually hurt the social benefit. In other words, the additional information on the server state is not very helpful for increasing equilibrium social benefit. As to the sensitivity of the equilibrium social benefit, we find that it is increasing with the arrival rate, vacation rate and service reward, which is intuitive. Now we pay more attention toFig. 7in which from a certain value ofh, the trend of the equilib- rium social benefit becomes smooth and steady. This is because that with the vacation rate increasing, on the one hand the vacation time reduces which benefits the social benefit, on the other hand since more people left for service, the service time increases which has a detrimental effect on the social benefit. When the vacation rate increases to a certain point, the po- sitive part counterbalances the negative part, therefore the equilibrium social benefit tends to reach a steady value.

5.2. An example

In daily life, it is common to see the queueing phenomenon, especially in the banks where lots of people are waiting for service. Now banks generally use the automatical calling system, from which the arriving customer takes a queue number.

The data series recorded by bank queueing machines are used to analyze the queueing