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We a give deterministic (sample path) proof of a result that extends the Pollaczek-Khintchine formula for a multiple vacation single-server queueing model. We also give a conservation law for the same system with multiple classes. Our results are completely rigorous and hold under weaker assumptions than those given in the literature. We do not make stochastic assumptions, so the results hold almost surely on every sample path of the stochastic process that describes the system evolution. The article is self contained in that it gives a brief review of necessary background material.

Consider a single-server queue with multiple vacations general arrival process and general-service times. The server works until all customers in the queue are served then takes a vacation; the server takes a second vacation if when he is back, there are no customers waiting, and so on, until he finds one or more waiting customers at which point he resumes service until all customers, including new arrivals, are served. A vacation is not initiated when there are customers in the system. Under the stochastic assumptions of stationarity and

This model has applications in communication systems and repair systems among others. Choudhury [

References also include Bisdikian [

Most analyses give the transform of the waiting time and queue length distribution and then invert these transforms to give explicit expressions. The reader is left with the task of understanding the details of the mathematical analysis. The contribution of this article is to give completely rigorous intuitive proofs that avoid transform methods. We accomplish this by using sample-path analysis. In other words, our proofs are intuitive, completely rigorous and hold pathwise in the sense that they are true on every realization of the stochastic process of interest. Our results turn out to be valid under weaker assumptions that those required in the literature.

This paper is organized as follows: in Section

In this section, we review a few preliminary results that are used in the proof of the main result. Our proof uses the sample path relation

We are given a deterministic sequence of time points,

With

There exists a sequence

In economic terms, Condition A says that all the cost associated with the

Suppose

We next consider two applications of

Let

The asymptotic average residual time is

This proof uses

An alternative proof of this result uses the simpler relation,

Consider any

For an

In the next section, we extend Pollaczek-Khintchine formula to systems with multiple vacations using

Consider any

Let

The two sequences

Note that the requirement that

Consider the G/G/1-FIFO multivacation model. Suppose that

Let

Recall that

Let

Equation (

Suppose that

Now we consider multiclass

Consider a bivariate sequence,

A single-server model with nonpreemptive

a single-server working at unit rate,

a set of

We also assume that the scheduling rules are

Consider the G/G/1 multivacation model with WCS rules that are nonpreemptive, regenerative and service time independent. Suppose that

Consider the multi-class

Consider the G/G/1 stable multi-class multiple vacation model with WCS rules that are non-preemptive, regenerative, and within each class, they are service time independent. Also, suppose

Let