Elsevier

Operations Research Letters

Volume 34, Issue 5, September 2006, Pages 531-538
Operations Research Letters

A workload-dependent M/G/1 queue under a two-stage service policy

https://doi.org/10.1016/j.orl.2005.08.002Get rights and content

Abstract

We consider an M/G/1 queueing system where the speed of the server depends on the amount of work present in the system. As a service policy, we adopt the Pλ,τM release policy in a dam model. By using the level crossing theory and solving the corresponding integral equations, we obtain the stationary distribution of the workload in the system explicitly.

Introduction

The Pλ,τM policy was introduced by Yeh [15] as a generalized release policy of the PλM policy of Faddy [11] for a dam with input formed by a Wiener process. Under the Pλ,τM policy, the release rate is zero until the level of water reaches λ(>0) and as soon as this occurs, water is released at rate M(>0) until the level of water reaches τ(0<τ<λ); once the water reaches level τ the release rate remains zero until level λ is reached again. Abdel-Hameed [1] considered the optimal control of a dam using Pλ,τM policies when the input process is a compound Poisson process with positive drift. Bae et al. [4] determined the long-run average cost per unit time under the Pλ,τM policy in a finite dam with a compound Poisson input where they assumed the initial releasing rate to be a constant a(0) instead of 0. Under the PλM policy, the stationary distribution of the workload in the M/G/1 queueing system with constant service speeds was derived in Bae et al. [3].

On the other hand, in queueing systems such as human-server systems the speed of the server often depends on the amount of work present. For example, Bertrand and Van Ooijen [7] described a production system where the speed of the server is relatively low when there is much work or when there is little work. See Bekker et al. [6] for more examples. In fact, models where the service speed is dependent on the workload originate from the studies of dams and storage processes. Dams with compound Poisson inputs and a general release rate function were studied by Asmussen [2], Cohen [10], Harrison and Resnick [12] and many others. For dams with more general input processes, see e.g. Brockwell et al. [8] and Kaspi et al. [13].

In this paper, we modify the Pλ,τM policy and introduce it as a two-stage service policy for the M/G/1 queueing system having workload-dependent service speeds; a server is initially idle and starts to serve, if a customer arrives, with service speed r1(·) depending on the amount of work present in the system. The customers arrive according to a Poisson process of rate ν(>0) and each customer brings a job consisting of an amount of work to be processed that is independently and identically distributed with a distribution function G and a mean m(>0). If the workload exceeds threshold λ(>0), the server changes his service speed to another workload-dependent speed r2(·) instantaneously and continues to follow that service speed until the workload level reaches τ(0<τ<λ). When the workload reaches level τ, the service speed is changed again to r1(·) instantaneously. The service speed r1(·) is kept until the level up-crosses λ again. We assume that r1(0)=0 and r1(·) and r2(·) are strictly positive, left continuous, and have a strictly positive right limit on (0,λ) and on (τ,), respectively. We also assume that 1/ri(x), for i=1,2, are integrable over any finite interval [a,b](0,). In particular, define R1(x)0x1r1(y)dy,0xλ,representing the time required for the workload process with the service speed r1(·) to become zero in the absence of any arrivals, starting with workload x. We assume that R1(x)< for all 0xλ, which implies that the workload process has an atom at level 0 (see [2, p. 288]). For the stability of the system, the speed functions must satisfy limsupxmν/ri(x)<1, for i=1,2.

Bar-Lev and Perry [5] had discussed a similar model for a dam to ours and derived integral equations whose solutions determine the stationary distribution of the dam content. They also demonstrated such a determination explicitly for the case of exponential jumps. In this paper, using a similar method as in Bae et al. [3], we derive and solve the integro-differential equation for the distribution of the workload at the exit time from (0,λ]. Together with the level crossing theory, it enables us to determine the explicit stationary distribution of the workload not only for the case of exponential jumps but also for the case of generally distributed jumps.

In Section 2, we determine the distribution of the excess amount over λ at an exit time from [0,λ] along with the distribution of the workload at the exit time from (0,λ]. Adopting the level crossing analysis carried out in Bar-Lev and Perry [5], the stationary distribution of the workload is derived in Section 3. In Section 4, as an example, we obtain an explicit distribution for the M/M/1 queueing system under the Pλ,τM policy with constant service speeds, which coincides with the result obtained in Bae et al. [3] in case τ=0.

Section snippets

The excess amount over λ at the exit time from [0,λ]

Let X(t) denote the workload of the system at time t under the service policy described in the previous section. Let X(0)=0 and define T0λ=inf{t>0|X(t)>λ},T0τ=inf{t>T0λ|X(t)=τ}and for n1, Tnλ=inf{t>Tn-1τ|X(t)>λ},Tnτ=inf{t>Tnλ|X(t)=τ},then {X(t),t0} is a delayed regenerative process having T0τ,T1τ,T2τ, as regeneration points. Fig. 1 depicts a typical realization of the workload process {X(t),t0}.

Since {X(t),t0} is non-Markovian, we decompose it into two Markov processes. Let {X1(t),t0} be

The stationary distribution

As illustrated in Fig. 1, we denote by C, C1, and C2 the cycles of the processes {X(t),t0}, {X1(t),t0}, and {X2(t),t0}, respectively. Then, obviously C=C1+C2.

Because {X(t),t0} and {Xi(t),t0} for i=1,2, are regenerative processes with finite mean cycles, each process has its stationary distribution function. Let Fi(x) be the stationary distribution function of {Xi(t),t0} for i=1,2, and let F(x) be that of {X(t),t0}. Then it follows thatF(x)=βF1(x)+(1-β)F2(x),where β=E[C1]/E[C]. Note that F

Example

In this section, we consider the special case of exponential jumps and positive constant service speeds, i.e., G(x)=1-e-x/m, r1(x)=r1 and r2(x)=r2, for all x>0. By the memoryless property of the exponential random variable, for all starting level 0<xλ, the probability P(l,x) in Lemma 2 is given by P(l,x)=1-e-l/m,l0.For i=1,2, let θi=1m-νri.SinceK*(x,y)=K1*(x,y)=νr1e-θ1(x-y),0<y<x<λand K2*(x,y)=νr2e-θ2(x-y),τ<y<x,we can obtain that Q(l,x)=νme-l/me-θ1(λ-x)-e-θ1λr1-νme-θ1λand Q(λ)=(r1-νm)e-θ1λr1-

Acknowledgement

We would like to thank an anonymous referee for valuable comments which significantly improved the representation of this paper.

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1

Supported by the Yeungnam University research grants in 2004.

2

Supported by the SRC/ERC program of MOST/KOSEF (R11-2000-073-00000).

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