A workload-dependent queue under a two-stage service policy
Introduction
The policy was introduced by Yeh [15] as a generalized release policy of the policy of Faddy [11] for a dam with input formed by a Wiener process. Under the policy, the release rate is zero until the level of water reaches and as soon as this occurs, water is released at rate until the level of water reaches ; once the water reaches level the release rate remains zero until level is reached again. Abdel-Hameed [1] considered the optimal control of a dam using 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 policy in a finite dam with a compound Poisson input where they assumed the initial releasing rate to be a constant instead of 0. Under the policy, the stationary distribution of the workload in the 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 policy and introduce it as a two-stage service policy for the queueing system having workload-dependent service speeds; a server is initially idle and starts to serve, if a customer arrives, with service speed depending on the amount of work present in the system. The customers arrive according to a Poisson process of rate 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 . If the workload exceeds threshold , the server changes his service speed to another workload-dependent speed instantaneously and continues to follow that service speed until the workload level reaches . When the workload reaches level , the service speed is changed again to instantaneously. The service speed is kept until the level up-crosses again. We assume that and and are strictly positive, left continuous, and have a strictly positive right limit on and on , respectively. We also assume that , for , are integrable over any finite interval . In particular, define representing the time required for the workload process with the service speed to become zero in the absence of any arrivals, starting with workload x. We assume that for all , 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 , for .
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 . 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 at an exit time from along with the distribution of the workload at the exit time from . 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 queueing system under the policy with constant service speeds, which coincides with the result obtained in Bae et al. [3] in case .
Section snippets
The excess amount over at the exit time from
Let denote the workload of the system at time t under the service policy described in the previous section. Let and define and for , then is a delayed regenerative process having as regeneration points. Fig. 1 depicts a typical realization of the workload process .
Since is non-Markovian, we decompose it into two Markov processes. Let be
The stationary distribution
As illustrated in Fig. 1, we denote by C, , and the cycles of the processes , , and , respectively. Then, obviously .
Because and for , are regenerative processes with finite mean cycles, each process has its stationary distribution function. Let be the stationary distribution function of for , and let be that of . Then it follows thatwhere . Note that
Example
In this section, we consider the special case of exponential jumps and positive constant service speeds, i.e., , and , for all . By the memoryless property of the exponential random variable, for all starting level , the probability in Lemma 2 is given by For , let Sinceand we can obtain that and
Acknowledgement
We would like to thank an anonymous referee for valuable comments which significantly improved the representation of this paper.
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Supported by the Yeungnam University research grants in 2004.
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Supported by the SRC/ERC program of MOST/KOSEF (R11-2000-073-00000).