1 queue with multiple working vacations

Performance Evaluation 63 (2006) 654–681

M/G/1 queue with multiple working vacations De-An Wua , Hideaki Takagib,∗ a

Doctoral Program in Policy and Planning Sciences, University of Tsukuba, 1-1-1 Tennoudai, Tsukuba-shi, Ibaraki 305-8573, Japan b Graduate School of Systems and Information Engineering, University of Tsukuba, 1-1-1 Tennoudai, Tsukuba-shi, Ibaraki 305-8573, Japan Received 8 September 2002; received in revised form 25 April 2005 Available online 19 July 2005

Abstract We study an M/G/1 queue with multiple vacations and exhaustive service discipline such that the server works with different service times rather than completely stopping service during a vacation. Both service times in a vacation and in a service period are generally distributed random variables. It is assumed that the Laplace–Stieltjes transform (LST) for the distribution of the vacation length is a rational function. We derive the distributions for the queue size and the system time for an arbitrary customer in the steady state. Several special cases, namely, exponentially distributed vacation lengths and/or exponentially distributed service times in a vacation, are considered. Finally some numerical examples are presented. © 2005 Elsevier B.V. All rights reserved. Keywords: M/G/1 queue; Multiple vacations; Working vacation

1. Introduction We consider an M/G/1 queue with multiple vacations such that the server works, say at a low rate, rather than completely stopping service during a vacation. Such a vacation is called a working vacation (WV) [10]. The server begins a working vacation of random length at the instant when the queue becomes empty. When a vacation ends, if there are customers in the queue, the server changes to another service rate, say a fast service rate, and the service interrupted at the end of the vacation restarts from the beginning; ∗

Corresponding author. Tel.: +81 29 853 5003; fax: +81 29 855 3849. E-mail address: [email protected] (H. Takagi).

0166-5316/$ – see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.peva.2005.05.005

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otherwise, another vacation is taken. The time interval between two successive vacations is called a service period [12, p. 107]. The length of a service period is zero if there are no customers waiting in the queue at the end of a vacation. The service times in the service period form an i.i.d. sequence of random variables having a general distribution function. The service times in the vacation are also i.i.d. with a general distribution function. The arrivals of customers occur according to a Poisson process. This queueing system is referred to as an M/G/1/WV queue in this paper. It is well known that the queueing system with server vacations is useful to model a system in which the server has additional task during a vacation. Thus, it has wide applicability in analyzing the performance of computer systems, data communication networks, and production systems. During the last two decades, the queueing systems with vacations have been studied extensively. The readers are referred to the surveys of Doshi [3,4] and the monograph of Takagi [12] as well as the references therein. In these studies, it is naturally assumed that the server stops service during the vacation. Recently, Servi and Finn [10] have analyzed an M/M/1/WV queue in which the server works at a different rate rather than completely stopping service during the vacation. The motivation of their study is that the M/M/1/WV queue can be used to approximate a multi-queue system whose service rate is one of two service speeds such that the fast speed mode cyclically moves from queue to queue with exhaustive service. They try to apply the M/M/1/WV queue to model a wavelength division multiplexing (WDM) optical access network using multiple wavelengths which can be reconfigured. The latter problem is addressed in [8] specifically. In their model, all random variables, i.e., interarrival times, service times in the service period, service times in the vacation, and the length of a vacation, are exponentially distributed. Therefore, it seems difficult to approximate the multi-queue system precisely. In this paper, we extend Servi and Finn’s M/M/1/WV model to an M/G/1/WV model. We assume that both service times in the service period and in the working vacation are generally distributed, and the Laplace–Stieltjes transform (LST) for the distribution of the vacation length is a rational function. We note that distributions with rational LSTs are dense in the set of all probability distributions on (0, ∞) [2, p. 86]. Thus, distributions with rational LSTs can approximate any distribution arbitrarily well. This model may be more suitable for approximating the multi-queue system. The rest of this paper is organized as follows. In Section 2 we describe the system and define the notation. Section 3 reviews some results about the transient behavior for an M/G/1 queue as a preliminary. In Section 4, an imbedded Markov chain that describes the queue size process in an M/G/1/WV queue is introduced, and the probability generating function (PGF) for the steady-state queue size is derived. Based on this, the PGF for the steady-state queue size at an arbitrary time is obtained in Section 5. The LST for the system time distribution of an arbitrary customer is analyzed in Section 6. Section 7 examines a special case in which the length of a vacation follows an exponential distribution. Another special situation in which the service time in a vacation is exponentially distributed is considered in Section 8. We show that the case of constant vacation length, whose LST is not a rational function, can be handled by our model when the service time in a vacation is exponentially distributed. Numerical examples are presented also in Sections 7 and 8. A summary of our study is given in Section 9. 2. System description and notation In our M/G/1/WV queue, the server begins a working vacation each time the queue becomes empty, and the length of a working vacation follows a distribution which has a rational LST. The service time in a working vacation is generally distributed. If the server returns from a working vacation to find no customers

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Fig. 1. Queue size and service process in an M/G/1/WV queueing system.

waiting, it begins another working vacation. If the server returns from a working vacation to find the queue not empty, it switches to a service period in which the service time follows a different distribution. In Fig. 1, we illustrate the queue size and the service process in the M/G/1/WV queue. It is noted that the service interrupted at the end of a vacation is lost and it is restarted with a different distribution at the beginning of the following service period. For the analysis of an M/G/1/WV queue, we introduce the following notation: λ v(t) V (t) H(t) B(t) φ(s) ψ(s) β(s) v vi h b bi

arrival rate probability density function (pdf) for the length of a working vacation
distribution function for the length of a working vacation, i.e., V (t) := 0t v(x) dx distribution function for the service time in a working vacation distribution function for the service time during a service period Laplace transform (LT) of v(t) LST of H(t) LST of B(t) mean of V (t) ith moment of V (t) (i = 2, 3, . . .) mean of H(t) mean of B(t) ith moment of B(t) (i = 2, 3, . . .)

3. Transient behavior of an M/G/1 queue Before starting our analysis, we review some results about the transient behavior for an M/G/1 queue, which will be used in studying the queue size process during a working vacation in an M/G/1/WV queue.

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Here we use the following notation: λ for the arrival rate, H(t) for the distribution function for the service time, and ψ(s) for the LST of H(t). Let Y (t) be the number of customers (queue size) in the M/G/1 queue at time t. Suppose that there is a departure at time t = −0 and Y (0) = i (≥0). We define the transition probability: Pi,j (t) := P{Y (t) = j|Y (0) = i};

t≥0

and its PGF: ∞ 

Γi (z, t) :=

Pi,j (t)zj .

(1)

j=0

The LT of Γi (z, t) γi (z, s) :=



∞

e−st Γi (z, t) dt;

0

i = 0, 1, 2, . . .

is given by [11, p. 74, Eq. (77); 12, p. 88, Eq. (7.73)]: γi (z, s) =

∗ zi+1 [1 − ψ(s + λ − λz)] + (z − 1)(s + λ − λz)ψ(s + λ − λz)Pi,0 (s) , (s + λ − λz)[z − ψ(s + λ − λz)]

(2)

where ∗ (s) := Pi,0



∞

0

e−st Pi,0 (t) dt =

[τ(s)]i s + λ − λτ(s)

and τ(s) is the root with the smallest absolute value of the equation: τ(s) = ψ[s + λ − λτ(s)].

(3)

Let Ni (t) denote the expected number of the customers served in the time interval (0, t] with initial queue size Y (0) = i (≥0). The LT of Ni (t) is given by [11, p. 78, Eq. (91)]: Ni∗ (s)



:=

∞



−st

e 0



ψ(s) s[τ(s)]i Ni (t) dt = 1− ; s[1 − ψ(s)] s + λ − λτ(s)

i = 0, 1, 2, . . .

(4)

4. Queue size in an embedded Markov chain We now analyze the M/G/1/WV queueing system described in Section 2. The queue size X(t) is the number of customers, including those waiting and those in service, in the system at time t. Notice that during a working vacation, the queue size process X(t) behaves exactly like the queue size process in an M/G/1 queue described in Section 3. This fact suggests that the queue size during each vacation in an M/G/1/WV queue can be analyzed by utilizing the transient solution for the queue size in an M/G/1 queue. Let us consider a discrete-time Markov chain C := {X(n) ; n = 0, 1, 2, . . .}, where X(n) denotes the queue size immediately after the epoch at which either a vacation starts or a service starts during a service

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period. Note that the epoch at which a service starts in a vacation is not a Markov point. In Fig. 1, the embedded Markov points are indicated by bullets on the time axis. For the time-homogeneous Markov chain C, the state transition probability is denoted by pij = P{X(n+1) = j|X(n) = i};

i, j = 0, 1, 2, . . .

Because the fact X(n) = 0 means that the nth Markov point is the beginning of a vacation, we have p0j =



∞

P0,j (t)v(t) dt;

0

j = 0, 1, 2, . . .

(5)

If X(n) > 0, the nth Markov point is the beginning of a service during a service period. Therefore, we have X(n+1) ≥ X(n) − 1, and pij =

⎧ ⎪ ⎨ ⎪ ⎩

(λt)j+1−i −λt e dB(t); j ≥ i − 1 (j + 1 − i)! j
∞

0

0;

i = 1, 2, . . .

(6)

Assuming that this Markov chain is ergodic, the limiting distribution: πi := lim P{X(n) = i}; n→∞

i = 0, 1, 2, . . .

satisfies the balance equations: πj = π0



∞

P0,j (t)v(t) dt +

0

j+1 

πi

i=1

 0

∞

(λt)j+1−i −λt e dB(t); (j + 1 − i)!

j = 0, 1, 2, . . .

(7)

and the normalization condition: ∞ 

πi = 1.

i=0

Let us introduce the PGF for {πi ; i = 0, 1, 2, . . .} by Π(z) :=

∞ 

πi zi .

i=0

Multiplying (7) by zj and summing over j = 0, 1, 2, . . ., we get Π(z) = where Ψ (z) :=

π0 [zΨ (z) − β(λ − λz)] , z − β(λ − λz)  0

∞

Γ0 (z, t)v(t) dt

is the PGF for the queue size at the end of a vacation and Γ0 (z, t) is defined in (1).

(8)

(9)

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To simplify the real integral (9) appearing in (8), we impose the following assumption. Assumption 1. The pdf v(t) for the length of a working vacation has a rational LT φ(s). We remark that distributions with rational LSTs are dense in the set of all probability distributions on (0, ∞) [2, p. 86]. Thus, distributions with rational LSTs can approximate any distribution arbitrarily well. For later use, we introduce the following lemma, where a function with exponential order r0 is defined as follows [7, p. 287]. Let f (t) be an almost piecewise continuous function [7, p. 241], and let it have the further property that there is a real number r0 such that lim f (t) e−rt = 0

when r > r0 ,

t→∞

and with the limit not existing when r < r0 . This limit is not necessarily satisfied if r = r0 . A function satisfying this condition is said to be of exponential order r0 . Lemma 1. Suppose that there is a function A(t) with exponential order 0, where t ∈ [0, ∞). Let a(s) :=



∞

0

e−st A(t) dt

for (s) > 0. If Assumption 1 holds, there are two small positive numbers  and c¯ such that the relation: 

∞

0

A(t)v(t) dt =

1 2πi



c+i∞

c−i∞

holds for  ≤ c ≤ c¯ , where i :=

a(s)φ(−s) ds

(10)

√ −1.




Proof. See [13, p. 144].

Remark 1. Eq. (10) may also be derived heuristically as follows:  0

∞

A(t)v(t) dt =



∞

0

v(t) dt

1 2πi



c+i∞

c−i∞

a(s) est ds =

1 2πi



c+i∞

c−i∞

a(s)φ(−s) ds.

(11)

This method is not mathematically rigorous, because the interchange of the order of integrations need to be justified. This is not always possible even when A(t) is of exponential order 0. However, if the conditions in Lemma 1 are satisfied we can prove that the expressions in (11) hold. On the other hand, the heuristic method has a merit of intelligibility. In the rest of the paper we use (11) frequently as long as the conditions in Lemma 1 are satisfied. Note that Γ0 (z, t) is of exponential order 0, since |Γ0 (z, t)| ≤ 1 for |z| ≤ 1. By Lemma 1, Eq. (9) can be represented by Ψ (z) =

1 2πi



c+i∞

c−i∞

γ0 (z, s)φ(−s) ds

for c ∈ [, c¯ ] and |z| ≤ 1. Substituting (2) into (12) yields: 1 Ψ (z) = 2πi



c+i∞

c−i∞



s+λ−λz z[1 − ψ(s + λ − λz)] + (z − 1)ψ(s + λ − λz) s+λ−λτ(s)

(s + λ − λz)[z − ψ(s + λ − λz)]

(12) 

φ(−s) ds.

(13)

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We can use the residue theorem to evaluate the integral in (13) over the contour consisting of the line (c + iR, c − iR) and a semicircle of radius R in the right-half plane which connects c − iR with c + iR counterclockwise. We choose c and R such that all the poles of φ(−s) are interior to this contour. The integration in (13) along the semicircle of radius R in the right-half plane which connects c − iR with c + iR counterclockwise tends to zero as R approaches infinity. Hence, the complex integral can be evaluated only at the poles of φ(−s). See [13, p. 125]for a detailed discussion. The PGF Π(z) in (8) is given in terms of the unknown constant π0 . In order to determine π0 , we use the following lemma. Lemma 2. If Assumption 1 holds, the order of differentiation and complex integration can be changed for c ∈ [, c¯ ] and |z| < 1 as follows: d dz



c+i∞

c−i∞



γ0 (z, s)φ(−s) ds =



c+i∞

c−i∞

∂ [γ0 (z, s)φ(−s)] ds ∂z

(14)

and similarly for higher derivatives.


Proof. See [13, p. 149].

Evaluating Ψ (z) in (13) at z = 1 gives 1 Ψ (1) = 2πi



c+i∞

φ(−s) ds = 1. s

c−i∞

(15)

Differentiating the right-hand side of (13) with respect to z and evaluating the result as z approaching 1 in the region of |z| < 1, we have lim z↑1

1 dΨ (z) = dz 2πi = =



∞

0



0

∞



c+i∞

c−i∞



λ ψ(s) s − 1− s2 s[1 − ψ(s)] s + λ − λτ(s)

1 v(t) dt 2πi



c+i∞



c−i∞



φ(−s) ds

λ ψ(s) s − 1− 2 s s[1 − ψ(s)] s + λ − λτ(s)



est ds

[λt − N0 (t)]v(t) dt = λv − N,

(16)

where we have used Lemma 2, Eq. (4) with i = 0, and N :=

 0

∞

1 N0 (t)v(t) dt = 2πi



c+i∞

c−i∞



ψ(s) s 1− φ(−s) ds s[1 − ψ(s)] s + λ − λτ(s)

(17)

is the expected number of customers served in a vacation. It can be shown that the complex integral in the third expression in (17) can be evaluated only at the poles of φ(−s). Recall that the expected number of customers served during the time interval (0, t] is not greater than the expected number of customers arriving during the same time interval, i.e., N0 (t) ≤ λt. Therefore, both N0 (t) and λt are of exponential order 0. The result in (16) means that the expected queue size at the end of a vacation is equal to the expected number of customers arriving during a vacation minus the expected number of customers served during that vacation. Thus, we have λv − N ≥ 0. By excluding the rare case λv − N = 0, let us assume that λv − N > 0 throughout the paper.

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We may determine the unknown constant π0 by recognizing that Π(1) = 1. Evaluating the right-hand side of (8) at z = 1 and using L’Hˆospital’s rule along with relations in (15) and (16), we obtain π0 =

1 − λb . 1 + λ(v − b) − N

(18)

Substituting this back into (8) yields: Π(z) =

(1 − λb)[zΨ (z) − β(λ − λz)] , [1 + λ(v − b) − N][z − β(λ − λz)]

(19)

where Ψ (z) and N are given in (13) and (17), respectively.

5. Queue size at an arbitrary time We next derive the PGF for the queue size in an M/G/1/MV queue at an arbitrary time. It is observed that the server has service periods and vacations alternately and that it is in one of the two periods at all times. If we consider the number of customers in the M/G/1/MV queue as the system state, it follows from the property of the Poisson arrival process and the assumption of exhaustive service that those points in time at which each vacation starts (i.e., at which each service period ends) are the regeneration points of the system. According to the renewal theory, if we select an epoch t randomly to observe the system, the probability that the server is in a vacation at time t is v pV := P{t ∈ vacation} = , (20) σ+v where σ is the mean of the length of a service period. Similarly, the probability that the server is in a service period at time t is σ pB := P{t ∈ service period} = . (21) σ+v Let S(t) be the distribution function for the length of a service period with LST: σ(s) :=



∞

e−st dS(t).

0

If there are i (≥0) customers in the system at the end of a vacation, the following service period behaves exactly like the busy period in an M/G/1 queue beginning with i customers. Therefore, if we denote by τB (s) the LST of the busy period distribution in an M/G/1 queue with arrival rate λ and service time distribution B(t), then σ(s) =

∞   i=0

0

∞

P0,i (t)v(t) dt[τB (s)]i =

 0

∞

Γ0 [τB (s), t]v(t) dt = Ψ [τB (s)],

where τB (s) is the root with the smallest absolute value of the equation τB (s) = β[s + λ − λτB (s)].

(22)

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Thus, we obtain



dΨ [τB (s)]  b = (λv − N), σ = −σ (0) = −  ds 1 − λb s=0

(23)

where we have used (16). Using (23) in (20) and (21), we get (1 − λb)v (λv − N)b , pB = . v − Nb v − Nb Since σ > 0 and λv − N > 0, it follows that pV =

(24)

1 − λb > 0.

(25)

Thus, we have 0 < (λv − N)b = λbv − Nb < v − Nb. Condition (25) means that the arrival rate is less than the service rate during a service period. It is a necessary condition for the stability of our system. By applying the theorem of total probability, the probability that the queue size X(t ) = j at an arbitrary time t is given by P{X(t ) = j} = pV P{X(t ) = j|t ∈ vacation} + (1 − δj0 )pB P{X(t ) = j|t ∈ service period}; j = 0, 1, 2, . . . ,

(26)

where δj0 is Kronecker’s delta. If we multiply (26) by zj and sum over j = 0, 1, 2, . . ., we obtain the PGF: Φ(z) = pV ΦV (z) + pB ΦB (z)

(27)

for the queue size at an arbitrary time, where Φ(z) :=

∞ 

P{X(t ) = j}zj ,

j=0

ΦV (z) :=

∞ 

P{X(t ) = j|t ∈ vacation}zj

(28)

P{X(t ) = j|t ∈ service period}zj .

(29)

j=0

and ΦB (z) :=

∞  j=1

To derive ΦV (z), note that the interval between an arbitrary time in a vacation and the starting time of that vacation corresponds to the deficit of the vacation time in the renewal theory. The pdf for the deficit is given by vˆ (t) :=

1 − V (t) v

(30)

D.-A. Wu, H. Takagi / Performance Evaluation 63 (2006) 654–681

and its LT is



∞

663



1 ∞ −st 1 − φ(s) e vˆ (t) dt = e [1 − V (t)] dt = . v 0 vs 0 Conditioning on the length of the deficit of a vacation time, we have ˆ := φ(s)

−st

P{X(t ) = j|t ∈ vacation} =






0

∞

P0,j (t)ˆv(t) dt;

(31)

j = 0, 1, 2, . . .

(32)

ˆ of vˆ (t) is a rational function if φ(s) It can be verified that 0t vˆ (x) dx is a proper distribution. The LT φ(s) is a rational function. Substituting (32) into (28) and using Lemma 1, we obtain 

ΦV (z) =

∞

0

Γ0 (z, t)ˆv(t) dt = Ψˆ (z),

where 

1 Ψˆ (z) := 2πi

⎡

(33)

s + λ − λz ⎤ s + λ − λτ(s) ⎥ ⎥ φ(−s) ˆ ds ⎦ (s + λ − λz)[z − ψ(s + λ − λz)]

c+i∞ ⎢ z[1 − ψ(s + λ − λz)] + (z − 1)ψ(s + λ − λz)

c−i∞

⎢ ⎣

(34) is the PGF for the queue size at an arbitrary time during a vacation. Here we note that c ∈ [, c¯ ] can be ˆ the same value as in (13). Again, the complex integral in (34) is evaluated only at the poles of φ(−s). It follows from Lemma 2 that if we differentiate (34) with respect to z for |z| < 1 and c ∈ [, c¯ ], the order of differentiation and complex integration can be changed. We next determine the PGF ΦB (z). Conditioning on the number of customers in the system at the epoch at which a service begins, and integrating with deficit of a service time, we obtain



P{X(t ) = j|t ∈ service period} =

j  i=1

πi 1 − π0

 0

∞

(λt)j−i −λt ˆ e dB(t); (j − i)!

j = 1, 2, . . . ,

(35)

where ˆ := 1 B(t) b

 0

t

[1 − B(x)] dx.

Substituting (35) into (29), rearranging terms, and using (8) and (18) gives π0 1 − π0

ΦB (z) =



Π(z) ˆ − λz) = − 1 β(λ π0

1 − λb b(λv − N)





z[Ψ (z) − 1][1 − β(λ − λz)] , (36) λ(1 − z)[z − β(λ − λz)]

where ˆ := β(s)

 0

∞

ˆ = e−st dB(t)

1 − β(s) . bs

Using relations (24), (33), and (36) in (27), we obtain



1 − λb z[Ψ (z) − 1][1 − β(λ − λz)] Φ(z) = vΨˆ (z) + . v − Nb λ(1 − z)[z − β(λ − λz)]

(37)

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Letting z = 1 in (37) recovers

1 − λb z[Ψ (z) − 1][1 − β(λ − λz)] Φ(1) = vΨˆ (1) + lim z→1 λ(1 − z)[z − β(λ − λz)] v − Nb

1 − λb [Ψ (1) − 1 + Ψ (1)]λb = v+ v − Nb λ(1 − λb)





1 − λb (λv − N)b = v+ v − Nb 1 − λb



= 1,

(38)

where we have used (15), L’Hˆospital’s rule, and Ψˆ (1) =



∞

1 − V (t) dt = 1. v

Γ (1, t)

0

Differentiating (37) and using L’Hˆospital’s rule four times, we have



1 − λb [λb2 + 2(1 − λb)b]Ψ (1) bΨ

(1) E[X] = vΨˆ (1) + + , v − Nb 2(1 − λb)2 2(1 − λb)

(39)

where Ψ (1) is given by (16), 1 Ψ (1) = πi





c+i∞



c−i∞

and 1 Ψˆ (1) = 2πi



c+i∞

λ2 λψ(s) s ψ(s) + λψ (s) − 1− + 3 2 2 s s [1 − ψ(s)] s[1 − ψ(s)] s + λ − λτ(s)



c−i∞

λ ψ(s) s − 1− 2 s s[1 − ψ(s)] s + λ − λτ(s)



φ(−s) ds

(40)



ˆ φ(−s) ds.

(41)

Differentiating (37) twice and using L’Hˆospital’s rule six times, we have

1 − λb E[X2 ] = v[Ψˆ (1) + Ψˆ

(1)] + v − Nb



× Ψ (1) + where 3 Ψ (3) (1) = 2πi





c+i∞

c−i∞



λb2 + 2(1 − λb)b λ[3λ2 b22 + 2(1 − λb)(3b2 + λb3 )] + 2(1 − λb)2 6(1 − λb)3 



b bΨ (3) (1) λb2 + 2(1 − λb)b

Ψ (1) + , + 2(1 − λb) 2(1 − λb)2 3(1 − λb)



2λ2 ψ(s) 2λ3 2λ[ψ(s) + λψ (s)] + − s4 s2 [1 − ψ(s)]2 s3 [1 − ψ(s)]

2ψ(s) + 2λψ (s)[1 + ψ(s) + λψ (s)] λ2 ψ

(s) + − s[1 − ψ(s)]3 s[1 − ψ(s)]2



(42)

s 1− s + λ − λτ(s)



× φ(−s) ds and 1 Ψˆ (1) = πi





c+i∞

c−i∞





(43)

λ2 λψ(s) s ψ(s) + λψ (s) − 1− + 3 2 2 s s [1 − ψ(s)] s[1 − ψ(s)] s + λ − λτ(s)



ˆ φ(−s) ds.

(44)

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We can show that if φ(s) is a rational function, the complex integrals in (16), (40), and (43) can be evaluated only at the poles of φ(−s), and the complex integrals in (41) and (44) can be evaluated only at ˆ the poles of φ(−s). Remark 2. If the server does not work during a working vacation, Φ(z) in (37) becomes [13, p. 154]: Φ(z) =

(1 − λb)[1 − φ(λ − λz)]β(λ − λz) . λv[β(λ − λz) − z]

(45)

This is the PGF for the queue size at an arbitrary time in an M/G/1 queue with multiple non-working vacations [12, p. 124, Eq. (2.19b)].

6. System time of an arbitrary customer We proceed to determine the total time T for an arbitrary customer to stay in the system (system time). We define the LST of the distribution function for the system time T by α(s) :=



∞

0

e−st dP{T ≤ t}.

To derive α(s), we use a theorem in [5] (the distributional form of Little’s law), which states as follows: Φ(z) = α(λ − λz),

(46)

if the following four conditions hold: (a) the arrivals are Poisson; (b) all arriving customers enter the system, and remain in the system until served; (c) the customers leave the system one at a time in the order of arrivals; (d) for any time t , the arrival process after time t and the time in the system of any customer arriving before t are independent. Obviously, these conditions are satisfied for our M/G/1/WV queue. Substituting z = 1 − s/λ in (37) gives

α(s) =



1 − λb (λ − s)[Ψ (1 − s/λ) − 1][1 − β(s)] vΨˆ (1 − s/λ) + . v − Nb s{λ[1 − β(s)] − s}

(47)

From (47) the mean E[T ] and the second moment E[T 2 ] of the system time T are given by

1 − λb E[X] [λb2 + 2(1 − λb)b]Ψ (1) bΨ

(1) = E[T ] = vΨˆ (1) + + λ (v − Nb)λ 2(1 − λb)2 2(1 − λb) and



(48)



2 2

1 − λb ˆ

(1) + λ[3λ b2 + 2(1 − λb)(3b2 + λb3 )]Ψ (1) v Ψ E[T ] = (v − Nb)λ2 6(1 − λb)3 2



[λb2 + 2(1 − λb)b]Ψ

(1) bΨ (3) (1) + + . 2(1 − λb)2 3(1 − λb)

(49)

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7. When the length of a vacation follows an exponential distribution Let us investigate the queue size and the system time when the length of a vacation follows an exponential distribution. In Section 7.1, the PGF for the queue size and the LST for the system time are derived. A numerical example is presented in Section 7.2. 7.1. Queue size and system time If the length of a vacation follows an exponential distribution with mean 1/η, then φ(s) = η/(s + η), thus φ(−s) = η/(η − s) has a pole of order 1 at s = η. By the residue theorem the PGF Φ(z) in (37) for the queue size X at an arbitrary time is reduced to η(1 − λb)[1 − ψ(η)]{η + λ[1 − τ(η)]} [1 − ψ(η)]{η + λ[1 − τ(η)]} − bλη[1 − τ(η)]ψ(η)

Φ(z) =

×



z[η(z) − 1][1 − β(λ − λz)] (z) + , λ(1 − z)[z − β(λ − λz)]

(50)

where (z) :=

η + λ(1 − z) η + λ[1 − τ(η)] . [η + λ(1 − z)][z − ψ(η + λ − λz)]

z[1 − ψ(η + λ − λz)] + (z − 1)ψ(η + λ − λz)

(51)

The mean E[X] and the variance Var[X] are given in Appendix A.1. In the same way as deriving (47), we obtain through (46) the LST of the distribution function for the arbitrary customer’s system time T as

 η(1 − λb)[1 − ψ(η)]{η + λ[1 − τ(η)]} (λ − s)[ηω(s) − 1][1 − β(s)] α(s)= ω(s) + , [1 − ψ(η)]{η + λ[1 − τ(η)]} − bλη[1 − τ(η)]ψ(η) s{λ[1 − β(s)] − s} (52) where ω(s) := (1 − s/λ) =

s+η (λ − s)[1 − ψ(s + η)] − sψ(s + η) η+λ[1−τ(η)]

(s + η){λ[1 − ψ(s + η)] − s} The mean E[T ] and the variance Var[T ] are also given in Appendix A.1.

.

(53)

Remark 3. Two special cases can be derived from (50) [13, p. 156]. (i) If the server is always on a working vacation, i.e., η → 0, (50) is reduced to (1 − λh)(1 − z)ψ(λ − λz) Φ(z) = , (54) ψ(λ − λz) − z where h is the mean of the service time in a working vacation. (ii) If the server never takes a vacation, i.e., η → ∞, (50) becomes Φ(z) =

(1 − λb)(1 − z)β(λ − λz) . β(λ − λz) − z

(55)

Each of (54) and (55) is referred to as the Pollaczek–Khinchin transform equation for an M/G/1 queue [12, p. 6, Eq. (1.18)].

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Remark 4. In the course of revising this paper, we have found that Kim et al. [6] analyze by decomposition approach the queue size distribution in an M/G/1/WV queue in which the length of a vacation follows an exponential distribution. Their expression in Eq. (10) of [6] for the PGF of the queue size is identical to (50). 7.2. Numerical example We illustrate the results obtained in Section 7.1 numerically. In our numerical example, we assume that the service time in a working vacation follows an Erlang-2 distribution with mean 1 for which the LST is given by



2 2 ψ(s) = . (56) s+2 The service time during a service period is assumed to follow an Erlang-3 distribution with mean 0.2 for which the LST is given by



3 15 . β(s) = s + 15 Combining (3) and (56) yields the equation:

τ(η) =

2 η + λ[1 − τ(η)] + 2

(57)

2

,

(58)

which we solve for τ(η) numerically. We plot the performance values by changing the arrival rate λ. In order to investigate the influence of the mean length v = 1/η of a vacation, we show the results for several values of v. Fig. 2 shows the means and the variances of the queue size and the system time. For comparison, the performance values for two M/Er /1 queues whose LSTs of the service time distributions are given by (56) and (57), respectively, are also plotted. It is observed that at large and small values of v our system behaves like an M/Er /1 queue in accordance with the limits given in (54) and (55).

8. When the service time in a vacation follows an exponential distribution In this section we consider the special situation in which the service time in a vacation follows an exponential distribution. In Section 8.1 the queue size and system time distributions are derived. A numerical example for an Erlang distributed vacation length is presented in Section 8.2. In Section 8.3 we show that the case of constant vacation length, whose LST is not a rational function, can be handled by our model as well. A numerical example of this case is also given. 8.1. Queue size and system time If the service time in a vacation follows an exponential distribution with rate ν, we have ν ψ(s) = . s+ν

(59)

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From (3) and (59), the LST τ(s) included in Φ(z) in (37) satisfies the equation: τ(s) =

ν , s + λ − λτ(s) + ν

(60)

Fig. 2. Performance when the service times in a working vacation and in a service period follow Erlang distribution with mean 1, order 2 and mean 0.5, order 3, respectively, and the length of a vacation is exponentially distributed with mean v. E1 and E2 denote two M/Er /1 queues and their service time distributions have mean 1, order 2 and mean 0.5, order 3, respectively. (a) Mean of the queue size. (b) Variance of the queue size. (c) Mean of the system time. (d) Variance of the system time.

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Fig. 2. (Continued ).

which may be written as 1 τ(s) = . s + λ − λτ(s) ν[1 − τ(s)]

(61)

Solving (60) yields τ(s) = τ1 (s) and τ2 (s), where τ1 (s) :=

s+λ+ν+



(s + λ + ν)2 − 4λν , 2λ

τ2 (s) :=

s+λ+ν−



(s + λ + ν)2 − 4λν . (62) 2λ

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We claim that τ2 (s) is the LST of the distribution function for the length of a busy period in a working vacation, since τ2 (s) is the only solution to the equation in (60) such that |τ(s)| ≤ 1. Now, by substituting (59) into (2) and using (61), we have ∗ zi+1 + (z − 1)νPi,0 (s) ; zs − (1 − z)(ν − λz)

γi (z, s) =

i = 0, 1, 2, . . . ,

(63)

where [τ2 (s)]i+1 . ν[1 − τ2 (s)]

∗ Pi,0 (s) =

(64)

From (4), the LT of the expected number of customers served in the time interval (0, t] with initial queue size X(0) = i (≥0) is given by Ni∗ (s) =

ν [τ2 (s)]i+1 − ; s2 s[1 − τ2 (s)]

i = 0, 1, 2, . . .

(65)

Thus, the expected number of customers served in a vacation, given in (17), is N=

1 2πi

est ds = ν





c+i∞

N0∗ (s)φ(−s) ds =

c−i∞ ∞

0



t−



t



∞

0

v(t) dt

1 2πi



c+i∞

c−i∞

ν τ2 (s) − s2 s[1 − τ2 (s)]

P0,0 (x) dx v(t) dt = ν(v − E1 ),

0



(66)

where we have used (64), and defined ⎡ ⎤    ∞⎢ t t t ⎥ ⎢ En := P0,0 (x)(dx)n ⎥ ··· ⎣ ⎦ v(t) dt; 0 0 0 0    

n = 1, 2, . . .

(67)

n

It can be verified that the bracketed part of the integrand in (67) is of exponential order 0. It follows from Lemma 1 that En can be rewritten as 1 En = 2πi



c+i∞

c−i∞

τ2 (s) φ(−s) ds. − τ2 (s)]

(68)

sn ν[1

We can show that the complex integral in (68) is evaluated only at the poles of φ(−s). Note that E1 is the sum of the mean lengths of those periods during which the queue is empty in a vacation. Substituting (63) and (66) into (37), we obtain the PGF for the queue size at an arbitrary time:

Φ(z) =



1 − λb z[Ψ (z) − 1][1 − β(λ − λz)] vΨˆ (z) + , v − νb(v − E1 ) λ(1 − z)[z − β(λ − λz)]

where 1 Ψ (z) = 2πi



c+i∞

c−i∞

⎡ ⎣

τ2 (s) z + (z − 1) 1−τ 2 (s)

zs − (1 − z)(ν − λz)

(69)

⎤ ⎦ φ(−s) ds

(70)

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and 

1 Ψˆ (z) = 2πi

671

⎡ c+i∞

c−i∞

τ2 (s) ⎤ 1 − τ2 (s) ⎥ ⎢ ⎥ˆ ⎣ zs − (1 − z)(ν − λz) ⎦ φ(−s) ds. z + (z − 1) ⎢

(71)

The mean E[X] and the variance Var[X] are given in Appendix A.2. In the same way as deriving (47), we obtain the LST of the distribution function for the arbitrary customer’s system time T as



(λ − s)[Ψ (1 − s/λ) − 1][1 − β(s)] 1 − λb vΨˆ (1 − s/λ) + . v − νb(v − E1 ) s{λ[1 − β(s)] − s}

α(s) =

(72)

The mean E[T ] and the variance Var[T ] are also given in Appendix A.2. Remark 5. It can be shown [13, p. 157] that if the length of a vacation follows an exponential distribution with mean 1/η and the service times in a vacation and that in a service period are exponentially distributed with rate ν and µ, respectively, Φ(z) in (69) is reduced to 

Φ(z) =

1− 1−

 −1 ν λ  1 − τ1 (η)µ 1 − τ1 (η)−1 µ ν λ 1 − τ1 (η)−1 z 1 − τ1 (η)µ z z µ

,

(73)

where τ1 (η) is defined in (62). The product form in (73) is the PGF for the queue size at an arbitrary time in an M/M/1/WV queue derived and interpreted by Servi and Finn [10]. 8.2. Numerical example for erlang distributed vacation length We illustrate the results obtained in Section 8.1 numerically. We assume that the service times in a service period and in a vacation are exponentially distributed with rate µ = 3.5 and ν = 1.5, respectively. The length of a vacation follows an Erlang distribution with pdf: 4 (−2t)/v te , v2

v(t) =

whose LST is given by

φ(s) =

2/v s + 2/v

2

.

(74)

As we stated earlier, En in (68) can be calculated by directly applying the residue theorem. Thus, we have 1 En = 2πi



c+i∞

c−i∞



τ2 (s) −4 d φ(−s) ds = 2 n s ν[1 − τ2 (s)] νv ds

vn−2 = n+1 (n + 1)(ν − λ)v2 − 2nv 2 ν



 τ2 (s)  n s [1 − τ2 (s)] s=2/v



(2 + λv)(λv + 2n + λnv) + νv[2 − 2λv + n(4 − 2λv) + (n + 1)νv]  . + (2/v + λ + ν)2 − 4λν From (74), it is easy to verify that the mean of v(t) is v, v2 = (3/2)v, and v3 = 3v3 .

(75)

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Fig. 3. Performance when the service times in a working vacation and in a service period are exponentially distributed with ν = 1.5, µ = 3.5, respectively, and the length of a vacation follows an Erlang distribution with order 2 and mean v. MM1 means an M/M/1 queue. (a) Mean of the queue size. (b) Variance of the queue size. (c) Mean of the system time. (d) Variance of the system time.

D.-A. Wu, H. Takagi / Performance Evaluation 63 (2006) 654–681

Fig. 3. (Continued ).

673

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Fig. 4. Performance when the service times in a working vacation and in a service period are exponentially distributed with ν = 1.5, µ = 3.5, respectively, and the length of a vacation is a constant v. MM1 means an M/M/1 queue. (a) Mean of the queue size. (b) Variance of the queue size. (c) Mean of the system time. (d) Variance of the system time.

D.-A. Wu, H. Takagi / Performance Evaluation 63 (2006) 654–681

Fig. 4. (Continued ).

675

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Fig. 3 shows the means and the variances of the queue size and the system time. For comparison, the performance values for two M/M/1 queues are also plotted. In this figure, we can observe the influence of the length of a vacation. Our queue behaves like an M/M/1 queue as the average vacation length becomes very small or very large. This result agrees with our intuition as well as with the limits in (54) and (55). 8.3. Numerical example for constant vacation length In this section we consider the case of constant vacation length when the service time in a vacation is exponentially distributed. Let the length of a vacation be a constant v, which means that φ(s) = e−vs . Consequently, vˆ (t) is the pdf of a uniform distribution defined on the interval [0, v]. We note that φ(s) is not a rational function. Thus, we cannot use Lemmas 1 and 2, because the conditions for these lemmas are not satisfied. In this case the queue size process during a vacation behaves exactly as in an M/M/1 queue. Therefore, Ψ (z) and Ψˆ (z) are given by 

1 v Ψˆ (z) = Γ0 (z, t) dt, v 0 respectively, where Γ0 (z, t) is the solution to the differential equation [9, p. 88] Ψ (z) = Γ0 (z, v)

and

∂ Γ0 (z, t) = (1 − z)[(ν − λz)Γ0 (z, t) − νP0,0 (t)] ∂t with the initial condition Γ0 (z, 0) = 1. The solution is given by z





(76)

(77) 

ν(1 − z) t Γ0 (z, t) = e (78) 1− P0,0 (x) e(−(1−z)(ν−λz)x)/z dx z 0 for 0 < z ≤ 1, where P0,0 (t) is given by means of its LT in (64) with i = 0. Combining (76) with (78), the expressions for Ψ (z) and Ψˆ (z) are given explicitly. It can be verified that the formulas for E[X], Var[X], E[T ], and Var[T ] in Appendix A.2 are still valid. Here En is given by ((1−z)(ν−λz)t)/z

En =



0

v



t



t

1 ··· P0,0 (x)(dx) = 2πi 0 0   n



c+i∞

c−i∞

τ2 (s) evs ds, − τ2 (s)]

sn ν[1

n = 1, 2, . . . ,

(79)

n

where the third expression follows from the inverse Laplace transform. We may use the algorithm in [1, p. 257] to calculate this inverse Laplace transform numerically. In the numerical example, we assume that the service times in a service period and in a vacation are exponentially distributed with rate µ = 3.5 and ν = 1.5, respectively. Fig. 4 shows the means and variances of the queue size and the system time. For comparison, the performance values for two M/M/1 queues are also plotted. It is observed that most features in this case are the same as in Fig. 3 except that the variances for the queue size and for the system time increase very slowly as λ increases.

9. Summary In this paper, we have analyzed an M/G/1/WV queue. It is assumed that the distribution function for the length of a vacation has a rational LST. The PGF for the queue size and the LST of the distribution function for the system time of an arbitrary customer have been derived by utilizing the transient solution

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for the queue size in an M/G/1 queue. Two special cases are examined, i.e., the case when the length of a vacation follows an exponential distribution and the case in which the service time in a vacation is exponentially distributed. The latter case with constant vacation length, whose LST is not a rational function, can also be handled by our model. Several numerical examples are presented. It is confirmed that when the vacation length is very large or very small an M/G/1/WV queue behaves like an M/G/1 queue. It is also observed that the variance in the length of a vacation influences strongly the variances of both queue size and system time.

Appendix A. Means and variances for the queue size and the system time in special cases A.1. Case of exponentially distributed vacation length • Mean queue size

E[X] =

λ 2η(1 − λb)[1 − ψ(η)](λ{τ(η) + (1 + ηb)[1 − τ(η)]ψ(η)} − η[1 − ψ(η)] − λ) × [λ2 [2b(1 + bη) − b2 η][1 − τ(η)][1 − ψ(η)]2 − 2bη2 [1 − τ(η)ψ(η)2 ](1 − λb) − [1 − ψ(η)](2λ + η(2 + ληb2 ) − 2[λ + η(2 + 3ηb − λb − 3ληb2 + ληb2 )]ψ(η) − {2λ(1 + ηb) − [2λ + η(2 + ληb2 )]ψ(η)}τ(η)) − 2λη2 b[1 − τ(η)](1 − λb)ψ (η)].

• Variance of the queue size

Var[X] =

1 (η + λ − (η + λ + bηλ)ψ(η) + λτ(η)(−1 + ψ(η) + bηψ(η)))2 ×

12η2 (−1

1 (λ(−3λ(ηλb2 (−1 + ψ(η))(η + λ − (2η + λ)ψ(η) + bλ)2 (−1 + ψ(η))2

+ τ(η)(−λ + (η + λ)ψ(η))) + 2(−1 + bλ)((1 + bη)(η + λ) + ψ(η)(−((1 + bη) × (3η + 2λ)) + (λ + η(2 + 3bη + bλ))ψ(η)) + bη2 λψ (η) − τ(η)(λ + bηλ − (η + 2λ + 2bηλ)ψ(η) + (1 + bη)(η + λ)ψ(η)2 + bη2 λψ (η))))2 + 2(η + λ − (η + λ + bηλ)ψ(η) + λτ(η)(−1 + ψ(η) + bηψ(η)))(6(1 + bη)(η + λ)(η + 2λ) × (−1 + bλ)2 + 3η2 λ3 b22 (−1 + ψ(η))2 (η + λ − (2η + λ) ×ψ(η) + τ(η)(−λ + (η + λ)ψ(η))) + 3ηλ(−1 + bλ)b2 (−1 + ψ(η))((η + λ)(3η + 2λ) − (3η + 2λ)2 ψ(η) + (8η2 + 7ηλ + 2λ2 )ψ(η)2 + 2η2 λψ (η) − τ(η)(−((η2 + 8ηλ + 4λ2 )ψ(η)) + (η + λ)(3η + 2λ)ψ(η)2 + λ(3η + 2λ + 2η2 ψ (η)))) + 2(−1 + bλ)(−(η2 λ2 b3 (−1 + ψ(η))2 (η + λ − (2η + λ)ψ(η) + τ(η)(−λ + (η + λ)ψ(η)))) + 3(−1 + bλ)((η2 (7 + 6bη) + 13η(1 + bη)λ + 6(1 + bη)λ2 )ψ(η)2 − (η2 (4 + 5bη) + 5η(1 + bη)λ

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+ 2(1 + bη)λ2 )ψ(η)3 + η2 λ((2 + 3bη)ψ (η) − 2bηλψ (η)2 − bηλψ

(η)) + ψ(η)(−((1 + bη)(η + 2λ)(4η + 3λ)) + η2 λ(−((2 + 7bη)ψ (η)) + bηλψ

(η))) + τ(η)(−((1 + bη)λ(η + 2λ)) + (bη3 − 7η(1 + bη)λ − 6(1 + bη)λ2 )ψ(η)2 + (1 + bη)(η + λ)(η + 2λ)ψ(η)3 + η2 λ(ψ (η)(−2 − bη + 2bηλψ (η)) + bηλψ

(η)) + ψ(η)(−η2 + 5η(1 + bη)λ + 6(1 + bη)λ2 + η2 λ((2 + 5bη)ψ (η) − bηλψ

(η))))))))). • Mean system time

E[T ] =

E[X] . λ

• Variance of the system time

Var[T ] =

1 (η + λ − (η + λ + bηλ)ψ(η) + λτ(η)(−1 + ψ(η) + bηψ(η)))2 ×

1 (3η2 λ3 b22 (−1 + ψ(η))2 (η + λ − (2η + λ)ψ(η) 12η2 λ(−1 + bλ)2 (−1 + ψ(η))2

+τ(η)(−λ + (η + λ)ψ(η)))(η + λ − (λ + 2bηλ)ψ(η) + τ(η)(−λ + (−η + λ + 2bηλ) × ψ(η))) − 12η2 λ(−1 + bλ)2 b2 (−1 + ψ(η))2 ((η + λ)(3η + λ)ψ(η) − η(3η + 2λ) × ψ(η)2 + λ2 τ(η)2 (−1 + ψ(η) − ηψ (η)) − (η + λ)(η + λ + ηλψ (η)) +τ(η)(2λ(η + λ) − 2λ(2η + λ)ψ(η) + η(η + 2λ)ψ(η)2 + ηλ(η + 2λ)ψ (η))) −4(−1 + bλ)(η2 λ2 b3 (−1 + ψ(η))2 (η + λ − (η + λ + bηλ)ψ(η) + λτ(η) × (−1 + ψ(η) + bηψ(η)))(η + λ − (2η + λ)ψ(η) + τ(η)(−λ + (η + λ)ψ(η))) + 3(−1 + bλ)((−1 + bη)(1 + bη)λ(η + λ)2 + 2(η3 (2 + bη) − 3η2 (1 + bη) × (−1 + 3bη)λ − 4η(−1 + b2 η2 )λ2 + (1 + bη)(2 + bη)λ3 )ψ(η)3 + (−2η3 (1 + bη) + η2 (−2 + bη(4 + 7bη))λ + 2η(−1 + b2 η2 )λ2 − (1 + bη)2 λ3 )ψ(η)4 + η2 λ(2(1 + bη)(η + λ)(−1 + bλ)ψ (η) + bηλ(2η + 2λ + bηλ) × ψ (η)2 + bηλ(η + λ)ψ

(η)) + τ(η)(2(−((−1 + bη)(1 + bη)λ2 (η + λ)) + ψ(η)(η3 + η2 λ + 4η(−1 + b2 η2 )λ2 + 2(−2 + bη)(1 + bη)λ3 + ψ(η)(−2η2 λ + 6λ3 − bη4 (1 + bλ) + 6ηλ2 (1 + bλ) − 2η3 (1 + bλ(2 + 3bλ)) + ψ(η)(η3 (1 + bη) + η2 (1 + 2bη(3 + 2bη))λ + 4η(−1 + b2 η2 )λ2 − 2(1 + bη)(2 + bη)λ3 − (1 + bη)λ(η + λ)(2bη2 − (1 + bη)λ)ψ(η)))) + η2 λ(η + 2λ − b2 η2 λ − 2b(1 + bη)λ2 + ψ(η)(−2η(1 + bη) + 2(−2 + bη(−2 + bη))λ + 4b(1 + bη)λ2 + (η(1 + 2bη) + (2 + bη(4 + bη))λ − 2b(1 + bη)λ2 )ψ(η)))ψ (η) + bη3 λ2 (−η − 2λ − bηλ

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+ (η + 2λ + 2bηλ)ψ(η))ψ (η)2 ) − bη3 λ2 (−1 + ψ(η))(−η − 2λ + (η + 2λ + 2bηλ)ψ(η))ψ

(η)) + ψ(η)2 (−2η3 + η2 (−7 + 5bη(2 + 3bη))λ + 12η(−1 + b2 η2 )λ2 − 6(1 + bη)λ3 + η2 λ(−2(η(1 + 3bη) + λ + 2bηλ − b(1 + bη)λ2 )ψ (η) + bηλ(η + λ +bηλ)ψ

(η))) + λψ(η)(−2(1 + bη)(η + λ)(−2λ + η(−2 + b(3η + λ))) + η2 (−4(η + λ)(−1 + b(−2η + λ + bηλ))ψ (η) − 2bηλ(η + λ + bηλ)ψ (η)2 − bηλ(2η + 2λ + bηλ)ψ

(η))) + λτ(η)2 (−2(1 + bη)(2 + bη)(η − λ)(η + λ)ψ(η)3 + (1 + bη)2 (η − λ)(η + λ)ψ(η)4 + ψ(η)2 (η2 (5 + 4bη) − 6(1 + bη)λ2 + η2 (1 + bη)λ × (−2(1 + bη − bλ)ψ (η) + bηλψ

(η))) + λ((−1 + b2 η2 )λ + η2 (ψ (η)(−2 + 2b(1 + bη)λ + bη(2 + bη)λψ (η)) + bηλψ

(η))) + ψ(η)(4λ2 − 2η(η + b(−1 + bη)λ2 ) + η2 λ(−2(1 + bη)ψ (η)(−2 + 2bλ + bηλψ (η)) − bη(2 + bη)λψ

(η))))))).

A.2. Case of exponentially distributed service time in a vacation • Mean queue size

E[X] =

1 {2λvb(1 − λb) + λb2 [λv − ν(v − E1 )] 2(1 − λb)[v − νb(v − E1 )] +(1 − λb)(1 − νb)[v2 (λ − ν) + 2νE2 ]}.

This is derived from (69) and using Ψ (1) =

1 2πi

1 Ψ (1) = 2πi





c+i∞

c−i∞



c+i∞

c−i∞





λ−ν τ2 (s) +ν 2 s sν[1 − τ2 (s)]



φ(−s) ds = λv − N = λv − ν(v − E1 ),



τ2 (s) τ2 (s) (λ − ν)2 2ν 2 + 2 − 2ν +2ν(λ − ν) 2 3 s s sν[1 − τ2 (s)] s ν[1−τ2 (s)]

×φ(−s) ds = (λ − ν)2 v2 + 2ν(v − E1 ) + 2ν(λ − ν)E2 and 1 Ψˆ (1) = 2πi 



c+i∞

c−i∞



λ−ν τ2 (s) +ν 2 s sν[1 − τ2 (s)] 





ˆ φ(−s) ds



c+i∞ λ − ν 1 ∞ τ2 (s) 1 = [1 − V (t)] dt +ν est ds v 0 2πi c−i∞ s2 sν[1 − τ2 (s)]    t (λ − ν)v2 νE2 1 ∞ [1 − V (t)] (λ − ν)t + ν + = P0,0 (x) dx dt = . v 0 2v v 0



680

D.-A. Wu, H. Takagi / Performance Evaluation 63 (2006) 654–681

• Variance of the queue size

Var[X] =

12(−1 +

−1 (3(−2bvλ(−1 + bλ) + λb2 (vλ + (−v + E1 )ν) + b(−v + E1 )ν)2

bλ)2 (v

+(−1 + bλ)(−1 + bν)(v2 (λ − ν) + 2E2 ν))2 + 2(−v + b(v − E1 )ν)(3λ3 b22 (vλ + (−v + E1 )ν) − 3λ(−1 + bλ)b2 (3vλ + v2 (λ − ν)2 + (−v + E1 + 2E2 (λ − ν))ν) +(−1 + bλ)(−2λ2 b3 (vλ + (−v + E1 )ν) + (−1 + bλ)(−2v3 (λ − ν)2 (−1 + bν) +3v2 (λ(1 + 2bλ) − ν(−1 + 3bλ + bν)) + 6(bvλ + ν(−2E3 (λ − ν)(−1 + bν) +E2 (−1 + 2bλ + bν))))))). This is derived from (69) and using E[X], 1 Ψ (3) (1) = 2πi

×



c+i∞



c−i∞

τ2 (s) 6ν sν[1 − τ2 (s)]

τ2 (s) s3 ν[1 − τ2 (s)]



−



+

6(λ − ν)3 ν(λ − ν) + 12 + 6ν(λ − ν)2 4 s s3

τ2 (s) 6ν − 6ν(λ − 2ν) s2 s2 ν[1 − τ2 (s)]



φ(−s) ds

= 6νE1 + (λ − ν)3 v3 + 6ν(λ − ν)v2 + 6ν(λ − ν)2 E3 − 6νv − 6ν(λ − 2ν)E2 = −6ν(v − E1 ) + (λ − ν)2 [v3 (λ − ν) + 6νE3 ] + 6ν(λ − ν)(v2 − E2 ) + 6ν2 E2 and 1 Ψˆ

(1) = 2πi



c+i∞



c−i∞

τ2 (s) (λ − ν)2 2ν 2 + 2 − 2ν 3 s s sν[1 − τ2 (s)]



τ2 (s) + 2ν(λ − ν) 2 s ν[1 − τ2 (s)]





1 (λ − ν)2 v3 ˆ × φ(−s) ds = + ν(v2 − 2E2 ) + 2ν(λ − ν)E3 . v 3 • Mean system time

E[T ] =

E[X] . λ

• Variance of the system time

Var[T ] =

1 12λ2 (−1

+

bλ)2 (v

+ b(−v + E1 )ν)2

(−3(−2bvλ(−1 + bλ) + λb2 (vλ + (−v + E1 )ν)

+ (−1 + bλ)(−1 + bν)(v2 (λ − ν) + 2E2 ν))2 + 2(−v + b(v − E1 )ν)(−3λ3 b22 (vλ + (−v + E1 )ν) + 3λ(−1 + bλ)b2 (2vλ + (λ − ν)(v2 (λ − ν) + 2E2 ν)) + 2(−1 + bλ) × (λ2 b3 (vλ + (−v + E1 )ν) + (−1 + bλ)(v3 (λ − ν)2 (−1 + bν) − 6ν(−(E3 (λ − ν) × (−1 + bν)) + E2 (−1 + bλ + bν)) − 3v2 (bλ2 − ν(−1 + bλ + bν)))))).

D.-A. Wu, H. Takagi / Performance Evaluation 63 (2006) 654–681

681

References [1] J. Abate, G.L. Choudhury, W. Whitt, An introduction to numerical transform inversion and its application to probability models, in: W.K. Grassmann (Ed.), Computational Probability, Kluwer Academic Publishers, Boston, MA, 2000, pp. 257–323. [2] S. Asmussen, Applied Probability and Queues, 2nd ed., Springer-Verlag, New York, 2003. [3] B.T. Doshi, Queueing systems with vacations—a survey, Queueing Syst. 1 (1) (1986) 29–66. [4] B.T. Doshi, Single server queues with vacations, in: H. Takagi (Ed.), Stochastic Analysis of Computer and Communication Systems, North-Holland, Amsterdam, 1991, pp. 217–226. [5] J. Keilson, L.D. Servi, A distributional form of Little’s law, Oper. Res. Lett. 7 (5) (1988) 223–227. [6] J.D. Kim, D.W. Choi, K.C. Chae, Analysis of queue-length distribution of the M/G/1 queue with working vacations (M/G/1/WV), Hawaii International Conference on Statistics and Related Fields, Honolulu, Hawaii, June 5–8, 2003. [7] W.R. LePage, Complex Variables and the Laplace Transform for Engineers, McGraw-Hill Book Company, New York, 1961 (Republished by Dover Publications, New York, 1980). [8] A. Narula-Tam, S.G. Finn, M. Medard, Analysis of reconfiguration in IP over WDM access networks, Proceedings of the Optical Fiber Communication Conference (OFC), 2001 pp. MN4.1–MN4.3. [9] T.L. Saaty, Elements of Queueing Theory with Applications, McGraw-Hill Book Company, New York, 1961 (Republished by Dover Publications, New York, 1983). [10] L.D. Servi, S.G. Finn, M/M/1 queues with working vacations (M/M/WV), Perform. Eval. 50 (1) (2002) 41–52. [11] L. Tak´acs, Introduction to the Theory of Queues, Oxford University Press, New York, 1962. [12] H. Takagi, Queueing Analysis: A Foundation of Performance Evaluation, vol. 1: Vacation and Priority Systems, Part 1, Elsevier Science Publishers B.V., Amsterdam, 1991. [13] D.-A. Wu, Queueing models for multimedia communication networks, PhD Dissertation, Institute of Policy and Planning Sciences, University of Tsukuba, January 2004. De-An Wu received B.S. degree in Management Sciences from the Changsha Railway University, China, in 1994. He also received M.A. degree in Economics and Ph.D. in Systems and Applied Mathematics from the University of Tsukuba, Japan, in 2000 and 2004, respectively. From 1994 to 1997 he was research and teaching assistant at the Changsha Railway University. He was a Postdoctoral Fellow at the Tohoku University from 2004 to 2005. He is now with the Tsukuba Advanced Research Alliance of the University of Tsukuba. His research interests include queueing theory, performance evaluation of communication systems, and image processing.

Hideaki Takagi is Professor in the School of Systems and Information Engineering at the University of Tsukuba, Japan. He received his B.S. and M.S. degrees in Physics from the University of Tokyo in 1972 and 1974, respectively. In 1974 he joined IBM Japan as a Systems Engineer. From 1979 to 1983, he studied at the University of California, Los Angeles, and received his Ph.D. degree in Computer Science. From 1983 to 1993, he was with IBM Research, Tokyo Research Laboratory. He moved to the University of Tsukuba in October 1993 as professor at the Institute of Policy and Planning Sciences. Prior to the current position, he was chair of the Doctoral Program in Policy and Planning Sciences in 1997– 1998, chair of the Institute in 2000–2001, and the Vice President of the University of Tsukuba in 2002– 2003. His research interests include enumerative combinatorics, probability theory, queueing theory and stochastic processes as applied to the performance evaluation of computer communication networks. He is the author of research monographs Analysis of Polling Systems (The MIT Press, 1986), and Queueing Analysis: A Foundation of Performance Evaluation, vols. 1–3 (Elsevier, 1991–1993). He is IEEE Fellow (1996) and IFIP Silver Core Holder (2001). He served as editors for IEEE Transactions on Communications (1986–1993) and IEEE/ACM Transactions on Networking (1992–1994). He currently serves as editors for Performance Evaluation (from 1984 onwards) and Queueing Systems (from 1988 onwards) journals.