Performance Evaluation 66 (2009) 356–367

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Performance Evaluation

journal homepage: www.elsevier.com/locate/peva

The GI/M/1 queue and the GI/Geo/1 queue both with singleworking vacationKyung C. Chae a,∗, Dae E. Lim a, Won S. Yang ba Department of Industrial & Systems Engineering, KAIST, Daejeon, 305–701, South Koreab Technology Strategy Research Division, ETRI, Daejeon, 305-700, South Korea

a r t i c l e i n f o

Article history:Received 26 March 2007Received in revised form 12 January 2009Accepted 31 January 2009Available online 7 February 2009

Keywords:Embedded Markov chainDiscrete-time queueWorking vacations

a b s t r a c t

We first consider the continuous-time GI/M/1 queue with single working vacation (SWV).During the SWV, the server works at a different rate rather than completely stoppingworking. We derive the steady-state distributions for the number of customers in thesystem both at arrival and arbitrary epochs, and for the FIFO sojourn time for an arbitrarycustomer. We then consider the discrete-time GI/Geo/1/SWV queue by contrasting it withthe GI/M/1/SWV queue.

© 2009 Elsevier B.V. All rights reserved.

1. Introduction

During the past two decades, queueing systems with server vacations have been studied extensively and applied tocomputer networks, communication systems and manufacturing systems [1,2].In the classical single server queues with server vacations, the server stops working during vacation periods. Suppose,

however, that a system can be staffed with a substitute server during the times the main server is taking vacations. Theservice rate of the substitute server is different from (and probably lower than) that of the main server. This is the notionof working vacations recently introduced by Servi and Finn [3]. Their work is motivated by the analysis of a reconfigurablewavelength-division multiplexing (WDM) optical access network.Servi and Finn [3] studied the M/M/1/MWV queue, where MWV stands for multiple working vacations. This study was

then extended to the GI/M/1/MWV queue by Baba [4] and to the M/G/1/MWV queue by Wu and Takagi [5]. Discrete-time versions of these also became available: Tian et al. [6] studied the Geo/Geo/1/MWV queue; Li et al. [7] studied theGI/Geo/1/MWV queue; and Yi et al. [8] studied the Geo/G/1/MWV queue. The MWV policy operates as follows. The serverbegins a working vacation when the system becomes empty. If the system is empty when the server returns from aworkingvacation, he begins another working vacation. Otherwise, he ends the vacation and changes the service rate back to theregular rate. Further studies related to the MWV policy can be found in [9–13].In this paper, we consider the single working vacation (SWV) policy. Under the SWV policy, the server takes only one

working vacation when the system becomes empty. Thus, if the system is empty when the server returns from a SWV, hestays in the system waiting for customers to arrive instead of taking another working vacation. Otherwise, he changes theservice rate back to the regular rate as under the MWV policy. Note that the SWV can be thought of as a post-processingtime during which the server works at a lower service rate rather than completely stopping working [1,2]. For example,suppose a machine with the policy that it has maintenance after one run of production while producing at a lower rate. In

∗ Corresponding author. Tel.: +82 42 869 2915; fax: +82 42 869 3110.E-mail address: kcchae@kaist.ac.kr (K.C. Chae).

0166-5316/$ – see front matter© 2009 Elsevier B.V. All rights reserved.doi:10.1016/j.peva.2009.01.005

K.C. Chae et al. / Performance Evaluation 66 (2009) 356–367 357

another case, an agent in call centers is required to do additional jobs after speaking with a customer [14]. The agent mayserve slowly the next customer while doing additional jobs (or cannot serve). Li and Tian [15] studied the Geo/Geo/1/SWVqueue. That study is extended in this paper to the GI/Geo/1/SWV queue. We also consider the GI/M/1/SWV queue.For the GI/M/1/SWV queue, we derive the steady-state distributions for the number of customers in the system both at

arrival and arbitrary epochs, and for the first-in first-out (FIFO) sojourn time for an arbitrary customer. We then considerthe discrete-time GI/Geo/1/SWV queue by contrasting it with the continuous-time GI/M/1/SWV queue. It is known thatdiscrete-time queues are more suitable for modeling digital communication systems [16,17].It is noteworthy that those readers who are unfamiliar with thematrix-geometric approach can understand the contents

of this paper well. We also demonstrate how to convert the discrete-time results into the corresponding continuous-timeresults.

2. Embedded Markov chain of the GI/M/1/SWV queue

We will follow the notation and assumptions in Baba [4] as much as possible. Let τn be the arrival epoch of the nthcustomer with τ0 = 0. Interarrival times Tn, n ≥ 1 are independent and identically distributed (iid) with a generaldistribution function, denoted by A(t) with a mean 1/λ and a Laplace Stieltjes transform (LST), denoted by A(s). The SWVtimes, the service times during a regular service period and the service times during a SWV are all exponentially distributedwith rates θ, µ and η, respectively.For the GI/M/1/SWV queue, let Q (t) be the number of customers in the system at time t and Qn = Q (τn − 0). Define

Jn =0, the nth customer arrives during a SWV,1, the nth customer arrives during a regular service period. (1)

Then the process (Qn, Jn), n ≥ 1 is an embedded Markov chain with the state space

Ω = (i, j) : i = 0, 1, 2, . . . ; j = 0, 1 (2)

Remark 1. The state (0, 1) does not exist in the GI/M/1/MWV queue [4].

In order to express the transition matrix of (Qn, Jn), n ≥ 1, let

P(i,j),(k,l) = P Qn+1 = k, Jn+1 = l | Qn = i, Jn = j , j, l = 0, 1, 0 ≤ k ≤ i+ 1,

bn =∫∞

0

(µt)ne−µt/n!

dA(t), n ≥ 0,

cn =∫∞

0e−θ t

(ηt)ne−ηt/n!

dA(t), n ≥ 0,

dn =∫∞

0

n∑j=0

∫ t

0θe−θx

(ηx)je−ηx

j!µ(t − x)n−j e−µ(t−x)

(n− j)!dxdA(t), n ≥ 0,

b′n =∫∞

0

∫ t

0

(µx)n−1e−µx

(n− 1)!µe−θ(t−x)dxdA(t), n ≥ 1,

c ′n =∫∞

0e−θ t

∫ t

0

(ηx)n−1e−ηx

(n− 1)!ηdxdA(t), n ≥ 1,

d′n =∫∞

0

n−1∑j=0

∫ t

0θe−θx

(ηx)je−ηx

j!

∫ t

x

µ(y− x)n−1−j e−µ(y−x)

(n− 1− j)!

×µe−θ(t−y)dydxdA(t), n ≥ 1.

Now we consider the transition probabilities. First, considering a regular service period, the transition from (i, 1) to(i+ 1− n, 1) occurs if there are n service completions during an interarrival time, where 0 ≤ n ≤ i. Thus, we have

P(i,1),(i+1−n,1) = bn, 0 ≤ n ≤ i. (3)

Second, considering a SWV period, the transition from (i, 0) to (i + 1 − n, 0) occurs if the SWV time is greater than aninterarrival time and there are n service completions during the interarrival time, where 0 ≤ n ≤ i. Thus, we have

P(i,0),(i+1−n,0) = cn, 0 ≤ n ≤ i. (4)

Third, the transition from (i, 0) to (i+ 1− n, 1) occurs if the ongoing SWV ends sometime during an interarrival time andthere are n service completions — j completions before the SWV ends and n − j after the SWV ends, where 0 ≤ j ≤ n and0 ≤ n ≤ i. Thus, we have

P(i,0),(i+1−n,1) = dn, 0 ≤ n ≤ i. (5)

358 K.C. Chae et al. / Performance Evaluation 66 (2009) 356–367

Note that Eqs. (3)–(5) are valid for theGI/M/1/MWVqueue aswell [4]. For theGI/M/1/SWV, however, we need to additionallyspecify P(i,1),(0,0) and P(i,0)(0,0). The transition from (i, 1) to (0, 0) occurs if there are i + 1 service completions during aninterarrival time and then the newly started SWV does not end during the remaining interarrival time. Thus, we have

P(i,1),(0,0) = b′i+1, i ≥ 0. (6)

Note that the quantity (µx)n−1 e−µxµ/(n − 1)! in b′n is not a Poisson mass function but an Erlang density function. Finally,the transition from (i, 0) to (0, 0) can occur in two different ways. First, this transition can occur if the SWV time is greaterthan an interarrival time and there are i+1 service completions during the interarrival time. The probability of this event isc ′i+1, i ≥ 0. Second, the transition from (i, 0) to (0, 0) can also occur if the ongoing SWVends sometime during an interarrivaltime but before there are i + 1 service completions, and the (i + 1)st service completion occurs during a regular serviceperiod, and then the newly started SWV does not end during the remaining interarrival time. The probability of this eventis d′i+1, i ≥ 0. Thus, we have

P(i,0),(0,0) = c ′i+1 + d′

i+1, i ≥ 0. (7)

Then, from Eqs. (3)–(7), we obtain

P(i,1),(0,1) = 1−i∑n=0

bn − b′i+1, i ≥ 0,

P(i,0),(0,1) = 1−i∑n=0

(cn + dn)− (c ′i+1 + d′

i+1), i ≥ 0.

Remark 2. For the GI/M/1/MWV queue [4], P(i,1),(0,1) = 0 and P(i,0),(0,1) = 0, i ≥ 0. (See Remark 1.)

We define generating functions as B(z) =∑∞

n=0 bnzn, C(z) =

∑∞

n=0 cnzn, D(z) =

∑∞

n=0 dnzn, B′(z) =

∑∞

n=1 b′nzn,

C ′(z) =∑∞

n=1 c′nzn, D′(z) =

∑∞

n=1 d′nzn, where 0 ≤ z ≤ 1.

Lemma 1.

B(z) = A(µ− µz),

C(z) = A(θ + η − ηz),D(z) = θ B(z)− C(z) / θ − (µ− η)(1− z) ,

B′(z) = µzB(z)− A(θ)/(θ − µ+ µz),

C ′(z) = ηzA(θ)− C(z)/(η − ηz),

D′(z) =θµz

θ − µ+ µz

B(z)

θ − (µ− η)(1− z)−

A(θ)η(1− z)

+

θµz · C(z)η(1− z) θ − (µ− η)(1− z)

.

Proof. The proof is long but straightforward. Two tools are necessary: the first is to switch the order of summations properlyand the second is the identity ex =

∑∞

n=0 xn/n!. For space considerations, we omit the details.

We now proceed to solve balance equations. Let

π(i,j) = limn→∞

P (Qn, Jn) = (i, j) , i = 0, 1, . . . ; j = 0, 1. (8)

Since one of the balance equations is redundant, we choose to ignore the one corresponding to state (0, 1). Then, the restare as follows:

π(i,0) =

∞∑n=0

cnπ(i−1+n,0), i ≥ 1, (9a)

π(i,1) =

∞∑n=0

dnπ(i−1+n,0) + bnπ(i−1+n,1)

, i ≥ 1, (9b)

π(0,0) =

∞∑n=1

(c ′n + d

′

n

)π(n−1,0) + b′nπ(n−1,1)

. (9c)

Our trial solution to Eq. (9a) is of the form

π(i,0) = π(0,0)r i1, 0 < r1 < 1, i ≥ 0. (10)

K.C. Chae et al. / Performance Evaluation 66 (2009) 356–367 359

Substituting Eq. (10) into Eq. (9a) leads to the equation

r1 =∞∑n=0

cnrn1 = C(r1) = A (θ + η − ηr1) . (11)

Substituting Eq. (10) into Eq. (9b) leads to

π(i,1) = π(0,0)r i−11 D(r1)+∞∑n=0

bnπ(i−1+n,1). (12)

For convenience, we define α as

α = θ/ θ − (µ− η)(1− r1) . (13)

Then, by Lemma 1 and Eq. (11), D(r1) in Eq. (12) can be expressed as

D(r1) = α B(r1)+ C(r1) = α

∞∑n=0

bnrn1 − r1

.

Thus, Eq. (12) can be rewritten as

π(i,1) + π(0,0)αr i1

=

∞∑n=0

bnπ(i−1+n,1) + π(0,0)αr i−1+n1

. (14)

Our trial solution to Eq. (14) is of the formπ(i,1) + π(0,0)αr i1

= π(0,0)βr i0, 0 < r0 < 1, i ≥ 0. (15)

Substituting Eq. (15) into Eq. (14) leads to the equation

r0 =∞∑n=0

bnrn0 = B(r0) = A(µ− µr0). (16)

Lemma 2. Under the stability condition that λ < µ and θ > 0, the equations z = C(z) and z = B(z) have unique roots in therange 0 < z < 1.

Proof. See Lemma 2 and Theorem 1 in Baba [4].

From Eqs. (10) and (15), and Lemma 2, we have

π(i,0) = π(0,0)r i1, i ≥ 0, (17a)

π(i,1) = π(0,0)(βr i0 − αr

i1

), i ≥ 0, (17b)

where r0 and r1 are the respective unique roots of z = B(z) and z = C(z) in the range 0 < z < 1. Substituting (17) intoEq. (9c), we have

1 =C ′(r1)+ D′(r1)

r−11 + βB

′(r0)r−10 − αB′(r1)r−11 .

Then, by using Lemma 1 and equations Eqs. (11) and (16), we can express β in Eq. (17b) as

β =θ − µ+ µr0

µr0 − A(θ)

[1+ η − αµ

η(1− r1)

r1 − A(θ)

]. (18)

Finally, π(0,0) in (17) can be obtained by 1 =∑∞

i=0

π(i,0) + π(i,1)

, and the result is

π(0,0) =

(β

1− r0+1− α1− r1

)−1. (19)

Remark 3. For the GI/M/1/MWV queue [4], β equals α as 0 = π(0,1) = π(0,0)(β − α). (See Remarks 1 and 2.)

Remark 4. The GI/M/1 queue with single exponential vacation in Chae et al. [18] corresponds to the case such that η→ 0.

360 K.C. Chae et al. / Performance Evaluation 66 (2009) 356–367

3. FIFO sojourn time distribution of the GI/M/1/SWV queue

Let fij(t) denote the conditional FIFO sojourn time density function of a customer who arrives at the system when itsstate is (i, j). Then, for i = 0, 1, . . ., and t ≥ 0, we have

fi1(t) =(µt)ie−µt

i!µ, (20a)

fi0(t) = e−θ t(ηt)ie−ηt

i!η +

∫ t

0θe−θx

i∑n=0

(ηx)ne−nx

n!µ(t − x)i−n e−µ(t−x)

(i− n)!µdx. (20b)

Note that fi1(t) is the density function of the Erlang random variable that represents the total service time for i + 1customers. The first term at the right side of Eq. (20b) represents the case that all i + 1 customers are served before theongoing SWV ends, and the second term represents the case that i or fewer are served before the ongoing SWV ends and therest after the SWV ends.Next step is to compute

∑∞

i=0 π(i,j)fij(t), j = 0, 1. This step is similar to the procedure of computing B(z) and D(z) sincefi1(t) and fi0(t) are similar to bn and dn, respectively. The results are

∞∑i=0

π(i,1)fi1(t) = π(0,0)βµe−(µ−µr0)t − αµe−(µ−µr1)t

,

∞∑i=0

π(i,0)fi0(t) = π(0,0)αµe−(µ−µr1)t + (η − αµ)e−(θ+η−ηr1)t

.

LetW and fw(t) denote the FIFO sojourn time of an arbitrary customer and its density function, then we have

fw(t) =∞∑i=0

1∑j=0

π(i,j)fij(t)

= π(0,0)βµe−(µ−µr0)t + (η − αµ)e−(θ+η−ηr1)t

, (21)

E[W ] = π(0,0)

β

µ(1− r0)2+

η − αµ

(θ + η − ηr1)2

. (22)

By using Eqs. (13) and (19), we can express fw(t) as

fw(t) = γ(µ− µr0)e−(µ−µr0)t

+ (1− γ )

(θ + η − ηr1)e−(θ+η−ηr1)t

,

where

γ =

(β

1− r0

)/(β

1− r0+1− α1− r1

).

Note that 0 < γ < 1 and thatW is a hyperexponential random variable.

Remark 5. fw(t) for the GI/M/1/MWV queue and the GI/M/1 queue with single exponential vacation can be obtained bysubstituting β = α and by taking the limit η→ 0, respectively. (See Remarks 3 and 4.)

4. The limiting distribution of Q (t) for the GI/M/1/SWV queue

Let L denote the steady-state system size at an arbitrary epoch, and let

Pn = P L = n = limt→∞

P Q (t) = n . (23)

We follow Baba’s [4] method of the semi-Markov process (SMP) to find Pn.Consider a new process (Z(t), K(t)), t ≥ 0, where Z(t) denotes the system size right after the most recent arrival

and K(t) equals 0 or 1 if the most recent arrival occurs during a SWV or during a regular service period, respectively.Then (Z(t), K(t)), t ≥ 0 is a SMP having (Qn + 1, Jn), n ≥ 1 for its embedded Markov chain. Note that π(i,j) =limn→∞ P (Qn + 1, Jn) = (i+ 1, j).Let TE denote the elapsed interarrival time at an arbitrary epoch in steady-state. The density function of TE is known as

λP T > t. Then, based on the relationship between Q (t), t ≥ 0 and (Z(t), K(t)), t ≥ 0, we have

Pn =1∑j=0

∞∑i=n−1

π(i,j)P i+ 1− n service completions during TE

K.C. Chae et al. / Performance Evaluation 66 (2009) 356–367 361

=

∞∑i=n−1

π(i,1)

∫∞

0

(µt)i+1−ne−µt

(i+ 1− n)!λP T > t dt +

∞∑i=n−1

π(i,0)

∫∞

0e−θ t

(ηt)i+1−ne−ηt

(i+ 1− n)!λP T > t dt

+

∞∑i=n−1

π(i,0)

∫∞

0

i+1−n∑k=0

∫ t

0θe−θx

(ηx)ke−ηx

k!×µ(t − x)i+1−n−k e−µ(t−x)

(i+ 1− n− k)!dxλP T > t dt. (24)

Substituting (17) into Eq. (24) and then carrying out computations as done in Baba’s [4] Eqs. (25)–(28), we obtain

Pn = λπ(0,0)

βrn−10

µ+(1− α)(1− r1)rn−11

θ + η − ηr1

, n ≥ 1, (25a)

P0 = 1−∞∑n=1

Pn = 1− λπ(0,0)

β

µ(1− r0)+

1− αθ + η − ηr1

. (25b)

Remark 6. Baba’s [4] Theorem 3 corresponds to the case that β = α. (See Remark 3.)

Remark 7. It can be shown that E[L] = λE[W ].

5. Preliminaries to the GI/Geo/1/SWV queue

Weadopt the early arrival system (EAS) [16,17]. Let the time-axis bemarked by 0, 1, . . .. According to the EAS, a potentialarrival takes place in (k, k+), k = 0, 1, . . ., and a potential departure occurs in (k−, k), k = 1, 2, . . .. It is assumed that aservice can only start at k+, k = 0, 1, . . ., and can only end at k−, k = 1, 2, . . .. Suppose that the length of the service thatstarts at k+ is l, l = 1, 2, . . ., then this service will end atm− wherem = k+ l.The GI/Geo/1/SWV queue is assumed to operate as follows. Interarrival times Tn, n ≥ 1 are iid random variables, and

have the distribution:

P Tn = i = ai, i = 1, 2, . . . ; E[Tn] = λ−1; A∗(z) =∞∑i=1

aiz i.

The service times during a regular service period Sn, n ≥ 1 are iid random variables, and follow a geometric distribution:

P Sn = l = µl−1µ, l = 1, 2, . . . ; 0 < µ < 1, µ = 1− µ.

During a regular service period, suppose that a customer departs the system in (k−, k) leaving behind no customers, thenthe single server begins a SWV at k, k = 1, 2, . . .. Suppose that the length of the SWV is l, l = 1, 2, . . ., then this SWV willend atmwherem = k+ l.

Remark 8. According to our assumption, the SWV that begins at k will continue even in the case a customer arrives in(k, k+), and it will last at least one time unit.

SWV times Vn, n ≥ 1 are iid random variables, and follow a geometric distribution:

P Vn = l = θ l−1θ, l = 1, 2, . . . ; 0 < θ < 1, θ = 1− θ.

The service times during a SWV periodS∗n , n ≥ 1

are iid random variables, and follow a geometric distribution:

PS∗n = l

= ηl−1η, l = 1, 2, . . . ; 0 < η < 1, η = 1− η.

Tn, n ≥ 1, Sn, n ≥ 1, Vn, n ≥ 1,S∗n , n ≥ 1

are mutually independent. It is assumed that λ < µ and θ < 1 for the

system to be stable (cf. Lemma 2).

6. Contrasting the GI/Geo/1/SWV queue with the GI/M/1/SWV queue

The analysis of the GI/Geo/1/SWV queue is quite parallel with that of the GI/M/1/SWV queue. For the GI/Geo/1/SWVqueue, let Q (t) be the system size at time t, t ≥ 0. Suppose that the nth customer arrives at the system in (k, k+). We defineQn as Qn = Q (k), k = n− 1, n, . . .. Then, the process (Qn, Jn), n ≥ 1 is an embedded Markov chain, where Jn and the statespace are as defined in Eqs. (1) and (2).

362 K.C. Chae et al. / Performance Evaluation 66 (2009) 356–367

In order to express the transition matrix of (Qn, Jn), n ≥ 1, let

bn =∞∑

i=max(1,n)

ai

(in

)µnµi−n,

cn =∞∑

i=max(1,n)

aiθ i(in

)ηnηi−n,

dn =∞∑

i=max(1,n)

aii∑j=1

θ j−1θ

m∑k=l

(jk

)ηkηj−k

(i− jn− k

)µn−kµi−j−n+k,

where l = max(0, n+ j− 1) andm = min(n, j),

b′n =∞∑i=n

aii∑j=n

(j− 1n− 1

)µn−1µj−nµ θ i−j,

c ′n =∞∑i=n

aiθ ii∑j=n

(j− 1n− 1

)ηn−1ηj−nη,

d′n =∞∑

i=max(2,n)

aii−1∑j=1

θ j−1θ

m∑k=l

(jk

)ηkηj−k

i∑g=j+1

(g − j− 1n− k− 1

)µn−k−1µg−j−n+kµθ i−g ,

where l = max(0, n+ j− g) andm = min(n− 1, j).Note that the quantity

(in

)µµi−n in bn is a binomial mass function whereas the quantity

(j−1n−1

)µn−1µj−nµ in b′n is a

negative binomial mass function. Further, the binomial and negative binomial mass functions correspond to the Poissonmass function and the Erlang density function in the continuous-time counterparts, respectively.Expressions for bn, cn, dn, b′n, c

′n and d

′n are different from their continuous-time counterparts but their meanings are

unchanged. Consequently, Eqs. (3)–(9) are valid for the GI/Geo/1/SWV queue as well.The generating functions in Lemma 1 now become

B(z) = A∗ (µz + µ) ,C(z) = A∗

θ (ηz + η)

,

D(z) =θ (ηz + η)

µz + µ− θ (ηz + η)B(z)− C(z) ,

B′(z) = µzB(z)− A∗(θ)

/(µz + µ− θ

),

C ′(z) = ηzA∗(θ)− C(z)

/ (η − ηz) ,

D′(z) =θµz (ηz + η)µz + µ− θ

B(z)

µz + µ− θ (ηz + η)−

A∗(θ)θ (η − ηz)

+

θµz(ηz + η)C(z)θ(η − ηz)

µz + µ− θ (ηz + η)

.The procedure of computing these generating functions is long but straightforward. A necessary tool is to switch theorder of summations properly. For example, when computing D′(z), we switch

∑∞

n=1∑∞

i=max(2,n)∑i−1j=1∑mk=l∑ig=j+1 into∑

∞

i=2∑i−1j=1∑ig=j+1

∑in=1

∑mk=l. Another necessary tool is the probability generating function (PGF) of the binomial random

variable. For example, when computing B(z), we usei∑n=0

(in

)(µz)i µi−n = (µz + µ)i .

The discrete-time counterparts of Eqs. (13) and (18) are as follows:

α = θ (ηr1 + η) /µr1 + µ− θ (ηr1 + η)

, (26a)

β =µr0 + µ− θµr0 − A∗(θ)

[1+ θη − αµ

θη(1− r1)

r1 − A∗(θ)

], (26b)

where r0 and r1 are the respective unique roots of z = B(z) and z = C(z) in the range 0 < z < 1. Note that the discrete-timecounterparts of Eqs. (11) and (16) are

r1 = C(r1) = A∗θ (ηr1 + η)

, (27a)

r0 = B(r0) = A∗ (µr0 + µ) . (27b)

K.C. Chae et al. / Performance Evaluation 66 (2009) 356–367 363

After all, (17) and (19) are also valid for the GI/Geo/1/SWV queue. That is,

π(i,0) = π(0,0)r i1, i ≥ 0, (28a)

π(i,1) = π(0,0)(βr i0 − αr

i1

), i ≥ 0, (28b)

π(0,0) =

(β

1− r0+1− α1− r1

)−1, (28c)

where α, β, r1, and r0 are as given in (26) and (27).

Remark 9. Li et al. [7] considered both the early arrival system(EAS) and the late arrival system(LAS). Under the EAS,Theorem 2 of Li et al. corresponds to the case that β = α. (See Remark 3.)

Remark 10. The GI/Geo/1 queue with single geometric vacation in Chae et al. [19] corresponds to the case such that η→ 0.(See Remark 4.)

LetWij denote the conditional FIFO sojourn time of a customer who arrives at the systemwhen its state is (i, j). Then, fori = 0, 1, . . ., and t = i+ 1, i+ 2, . . ., we have

P Wi1 = t =(t − 1i

)µiµt−1−iµ, (29a)

P Wi0 = t = θ t−1(t − 1i

)ηiηt−1−iη +

t−1∑j=1

θ j−1θ

m∑k=l

(jk

)ηkηj−k

(t − j− 1i− k

)µi−kµt−j−1−i+kµ, (29b)

where k = max(0, i + j − t + 1) and m = min(i, j). Note thatWi1 is a negative binomial random variable that representsthe total service time for i+ 1 customers. The first term at the right side of (29b) represents the case that all i+ 1 customersare served before the ongoing SWV ends, and the second term represents the case thatm or less are served before the SWVends and the rest after the SWV ends.Next step is to compute

∑t−1i=0 π(i,j)P

Wij = t

, j = 0, 1. This step is similar to the procedure of computing B(z) andD(z),

and the results aret−1∑i=0

π(i,1)P Wi1 = t = π(0,0)βµ (µr0 + µ)t−1 − αµ (µr1 + µ)t−1

,

t−1∑i=0

π(i,0)P Wi0 = t = π(0,0)[αµ (µr1 + µ)t−1 + (η − αµ)

θ (ηr1 + η)

t−1].

LetW denote the FIFO sojourn time of an arbitrary customer, then we have

P W = t =t−1∑i=0

1∑j=0

π(i,j)PWij = t

= π(0,0)

[βµ (µr0 + µ)t−1 + (η − αµ)

θ (ηr1 + η)

t−1], (30)

E[W ] = π(0,0)

[β

µ(1− r0)2+

η − αµ1− θ (ηr1 + η)

2]. (31)

Remark 11. Li et al. [7] considered the FIFO queue waiting time instead of the FIFO sojourn time. However, it can be shownthat the FIFO sojourn time for the GI/Geo/1/MWV queue corresponds to the case that β = α. (See Remarks 5 and 9.)

Let L and Pn denote the steady-state system size and its mass function at an arbitrary epoch. A characteristic of discrete-time queues is that the system size do not change during (k+, (k + 1)−), k = 0, 1, . . .. Thus, for convenience, we definePn as

Pn = P L = n = limk→∞

PQ (k+) = n

. (32)

Consider a new process (Z(t), K(t)) , t ≥ 0, where Z(t) denotes the system size right after the most recent arrivaland K(t) equals 0 or 1 if the most recent arrival occurs during a SWV or during a regular service period. Suppose thatthe most recent arrival is the nth arrival and that it occurs in (k, k+). Then, Qn = Q (k) and Qn + 1 = Q (k+) =Z(k+). Thus, (Z(t), K(t)) , t ≥ 0 is a SMP having (Qn + 1, Jn) , n ≥ 1 for its embedded Markov chain. Note that π(i,j) =limn→∞ P (Qn + 1, Jn) = (i+ 1, j).

364 K.C. Chae et al. / Performance Evaluation 66 (2009) 356–367

Fig. 1. The E[W ] of the E3/M/1 over the mean length of a vacation.

Let TE denote the elapsed interarrival time at k+ in steady-state, i.e., when k→∞. The mass function of TE is known asP TE = l = λP T > l. Then, based on the relationship between Q (t), t ≥ 0 and (Z(t), K(t)) , t ≥ 0, we have

Pn =1∑j=0

∞∑i=n−1

π(i,j)P i+ 1− n service completions during TE

=

∞∑i=n−1

π(i,1)

∞∑j=i+1−n

λP T > j(

ji+ 1− n

)µi+1−nµj−i−1+n

+

∞∑i=n−1

π(i,0)

∞∑j=i+1−n

λP T > j θ j(

ji+ 1− n

)ηi+1−nηj−i−1+n +

∞∑i=n−1

π(i,0)

∞∑j=i+1−n

λP T > j

×

j∑h=1

θh−1θ

m∑k=l

(hk

)ηkηh−k

(j− h

i+ 1− n− k

)µi+1−n−kµj−h−i−1+n+k, (33)

where l = max(0, h+ i+ 1− j− n) andm = min(h, i+ 1− n).Substituting (28) into (33) and then carrying out computations, we finally obtain

Pn = λπ(0,0)

βrn−10

µ+(1− α)(1− r1)rn−11

1− θ (ηr1 + η)

, n ≥ 1, (34a)

P0 = 1− λπ(0,0)

β

µ(1− r0)+

1− α1− θ (ηr1 + η)

. (34b)

Remark 12. It can be shown that E[L] = λE[W ].

Remark 13. Li et al. [7] did not consider Pn, n ≥ 0. However, it can be shown that Pn for the GI/Geo/1/MWV queuecorresponds to the case that β = α. (See Remarks 9 and 11.)

7. Concluding remarks

In general, it is easier to analyze continuous-time queues than corresponding discrete-time queues. Thus, we firstanalyzed the continuous-time GI/M/1/SWV queue. Then, by analogy with the results for the GI/M/1/SWV, we obtained thecorresponding results for the discrete-time GI/Geo/1/SWV queue.

K.C. Chae et al. / Performance Evaluation 66 (2009) 356–367 365

Fig. 2. The E[W ] of the H2/M/1 over the mean length of a vacation.

Takagi states in the preface of [17] that ‘‘it is by no means possible to convert all the results for continuous-time systemsinto corresponding results for discrete-time systems (conversion in the opposite direction is easy).’’ We now demonstratehow to convert discrete-time results into corresponding continuous-time results.We use a two-step procedure. In the first step, we reduce the size of a slot from 1 to 1/k, k > 1. We also reduce λ, θ, µ

and η to λ/k, θ/k, µ/k and η/k, respectively. In the discrete-time queues, the length of an interarrival time is expressedin terms of the number of slots that it spans. Thus, T slots of size 1 corresponds to kT slots of size 1/k. In the second step,we take the limit k → ∞. Note that the interarrival time is a discrete random variable in the first step but it becomes acontinuous random variable in the second step. Let dt denote limk→∞ 1/k.For example, we have

C(z) = A∗θ (ηz + η)

= E

[(1− θ)(1− η + ηz)T

]for the GI/Geo/1/SWV queue. The corresponding quantity for the GI/M/1/SWV is then obtained as

limk→∞

E

[(1−

θ

k

)(1−

η − ηzk

)kT]= E

[e−θTe−(η−ηz)T

]= A(θ + η − ηz).

The next example is to convert Eq. (34a) into Eq. (25a). The factor1− θ (ηr1 + η)

in Eq. (34a) can be expressed as

[θ 1− (η − ηr1) + (η − ηr1)]. Then, taking the limit k→∞ to

λ

kπ(0,0)

βrn−10µ

k

+(1− α)(1− r1)rn−11θk

(1− η−ηr1

k

)+

η−ηr1k

leads to Eq. (25a).As the last example, we convert (30) into Eq. (21) as follows:

limk→∞

π(0,0)

[βµ

k

(1− µ−µr0

k

)kt(1− µ−µr0

k

) + η − αµk

(1− θ

k

) (1− η−ηr1

k

)kt(1− θ

k

) (1− η−ηr1

k

) ]= π(0,0)

βµdte−(µ−µr0)t + (η − αµ)dte−(θ+η−ηr1)t

= fw(t)dt.

8. Numerical examples

The results of some numerical examples are depicted in three figures. We consider three types of interarrival timedistributions: Erlang in Fig. 1, hyperexponential in Fig. 2, and deterministic in Fig. 3. The horizontal axis of each figurerepresents the mean length of a vacation, θ−1, and the vertical axis represents the mean waiting times, E[W ].

366 K.C. Chae et al. / Performance Evaluation 66 (2009) 356–367

Fig. 3. The E[W ] of the D/M/1 over the mean length of a vacation.

Essentially, we compare the relative performance of three vacation policies: the SWV, the MWV, and the (non-working)single vacation denoted by SV. E[W ]s for these three policies are depicted in the same figure.In every example, the regular service rateµ is set to 1 and the arrival rate to 3/4. On the other hand, we use two different

values for the service rate during a working vacation: we set η = 1/4 in part (a) and η = 3/4 in part (b) of each figure.Specific interarrival time distributions used are as follows. In Fig. 1, we use Erlang (3, 9/4). That is, an interarrival time

is the sum of three independent exponential random variables having a common rate 9/4. The density function used inFig. 2 is

pλ1e−λ1t + (1− p)λ2e−λ2t ,

where p = 0.25, λ1 = 1.5 and λ2 = 0.5. In Fig. 3, the interarrival times are deterministically 4/3.The overall results are intuitively appealing. First, the SWV policy having η > 0 outperforms the SV policy having η = 0.

Second, the SWV policy outperforms the MWV policy as we set µ > η. Finally, it is possible for the SV policy to outperformthe MWV policy when the values of η and θ are relatively low.

References

[1] B.T. Doshi, Queueing systems with vacations — a survey, Queueing Syst. 1 (1986) 29–66.[2] H. Takagi, Queueing Analysis, vol. 1, North-Holland, Amsterdam, 1991.[3] L.D. Servi, S.G. Finn, M/M/1 queues with working vacations (M/M/1/WV), Perform. Eval. 50 (2002) 41–52.[4] Y. Baba, Analysis of a GI/M/1 queue with multiple working vacations, Oper. Res. Lett. 33 (2005) 201–209.[5] D. Wu, H. Takagi, M/G/1 queue with multiple working vacations, Perform. Eval. 63 (2006) 654–681.[6] N.S. Tian, Z.Y. Ma, M.X. Liu, The discrete-time Geom/Geom/1 queue with multiple working vacations, Appl. Math. Modelling (2007)doi:10.1016/j.apm.2007.10.005.

[7] J.H. Li, N.S. Tian, W.Y. Liu, Discrete-time GI/Geo/1 queue with multiple working vacations, Queueing Syst. 56 (2007) 53–63.[8] X.W. Yi, J.D. Kim, D.W. Choi, K.C. Chae, The Geo/G/1 queue with disasters and multiple working vacations, Stoch. Models 23 (2007) 537–549.[9] A.D. Banik, U.C. Gupta, S.S. Pathak, On theGI/M/1/N queuewithmultipleworking vacations – analytic anaylsis and computation, Appl.Math.Modelling31 (2007) 1701–1710.

[10] M. Jain, P.K. Agrawal, M/Ek/1 queueing systems with working vacation, Qual. Tech. Quantitative Management 4 (2007) 455–470.[11] J.H. Li, N.S. Tian, The discrete-time GI/Geo/1 queue with working vacations and vacation interruption, Appl. Math. Comput. 185 (2007) 1–10.[12] J.H. Li, N.S. Tian, Z.Y. Ma, Performance analysis of GI/M/1 queue with working vacations and vacation interruption, Appl. Math Modelling (2007)

doi:10.1016/j.apm.2007.09.017.[13] W.Y. Liu, X.L. Xu, N.S. Tian, Stochastic decomposition in the M/M/1 queue with working vacations, Oper. Res. Lett. 35 (2007) 596–600.[14] K. Kawanishi, QBD approximations of a call center queueing model with general patience distribution, Comput. Oper. Res. 35 (2008) 2463–2481.[15] J.H. Li, N.S. Tian, Analysis of the discrete-time Geo/Geo/1 queuewith single working vacation, Quality Tech. QuantitativeManagement 5 (2008) 77–89.[16] J. Hunter, Mathematical Techniques of Applied Probability, vol. 2, Academic Press, New York, 1993.[17] H. Takagi, Queueing Analysis, vol. 3, North-Holland, Amsterdam, 1993.[18] K.C. Chae, S.M. Lee, H.W. Lee, On stochastic decomposition in the GI/M/1 queue with single exponential vacation, Oper. Res. Lett. 34 (2006) 706–712.[19] K.C. Chae, S.M. Lee, S.H. Lee, The discrete-time GI/Geo/1 queue with single geometric vacation, Qual. Tech. Quantitative Management 5 (2008) 21–31.

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Kyung C. Chae received his master’s degree and Ph.D. from the Ohio State University. Currently, he is a professor of Industrial &Systems Engineering at KAIST. His research interest includes stochastic modeling and queueing theory.

Dae E. Lim received his master’s degree in industrial engineering from KAIST, Daejeon, Korea, in 2006. Currently, he is a Ph.D.student in Industrial & Systems Engineering, KAIST. His research interests are stochastic modeling and queueing theory.

Won S. Yang received his master’s degree and Ph.D. from KAIST, Daejeon, Korea. Currently, he is working as a senior researcherin Technology Strategy Research Division, ETRI (Electronics and Telecommunication Research Institute), Daejeon, Korea. His mainresearch interests are in stochastic modeling, queueing theory and the performance analysis of communication networks.