Sample-Path Analysis of Single-Server Queue with Multiple Vacations

Abstract

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.

1. Introduction

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 𝑖.𝑖.𝑑. inter-arrival and service times, it is known that the mean customer delay in the queue is the sum of two components: mean queue delay and mean vacation time.

This model has applications in communication systems and repair systems among others. Choudhury [1] considers a batch arrival queue with a single vacation where a server takes exactly one vacation after the end of each busy period. This model has applications in manufacturing systems of job-shop type. The paper derives the steady-state distribution of queue length, busy period, and unfinished work using Laplace-Stieltjes transformation approach. Boxma and Groenendijk [2] give (pseudo) conservation laws for multiqueue single server systems with cyclic service. Their work is based on well-known stochastic decomposition results. Green and Stidham Jr. [3] use a sample-path approach (similar to ours) to address conservation laws and their applications to scheduling and fluid systems. They use the achievable region method to solve stochastic problems utilizing mathematical programming formulations.

References also include Bisdikian [4] who presents a simple method for obtaining conservation laws for single-server queues with batch arrivals and multiple classes using the ASTA (El-Taha [5]) property. Relevant references also include Green and Stidham [3], Boxma and Groenendijk [2], and Choudhury [1]. There is a vast literature on queues with vacations, and conservation laws. The reader may consult Takagi [6] and references therein.

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 2, we give preliminary background results that are used in proving main results. In particular, we review 𝐻=𝜆𝐺 relation and give sample-path proofs for the residual service time and Pollaczeck-Khintchine formula for M/G/1 queues. In Section 3, we focus on systems with multiple vacations and give the main result which relates virtual delay at any given time with systems actual delay and vacation times. In Section 4, we give a sample path proof of the conservation law and consider a few special cases.

2. Preliminary and Background Results

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 𝐻=𝜆G which is a generalization of the well-known Little's formula.

We are given a deterministic sequence of time points, {𝑇_𝑘,𝑘≥1}, with 0≤𝑇_𝑘≤𝑇_𝑘+1<∞, 𝑘≥1, and we define 𝑁(𝑡)∶=max{𝑘≥1∶𝑇_𝑘≤𝑡}, 𝑡≥0, so that 𝑁(𝑡) is the number of points in [0,𝑡]. We assume that 𝑇_𝑘→∞ as 𝑘→∞, so that there are only a finite number of events in any finite time interval (𝑁(𝑡)<∞ for all 𝑡≥0), and we note that 𝑁(𝑡)→∞ as 𝑡→∞, since 𝑇_𝑘<∞ for all 𝑘≥1. Associated with each time point 𝑇_𝑘, there is a function 𝑓_𝑘∶[0,∞)→[0,∞). The bivariate sequence {(𝑇_𝑘,𝑓_𝑘(⋅)),𝑘≥1} constitutes the basic data, in terms of which the behavior of the system is described. We assume that 𝑓_𝑘(𝑡) is Lebesgue integrable on 𝑡∈[0,∞), for each 𝑘≥1.

With 𝐻(𝑡) and 𝐺_𝑘 defined by 𝐻(𝑡)=∑^∞𝑘=1𝑓_𝑘(𝑡) and 𝐺_𝑘=∫^∞0𝑓_𝑘(𝑡)𝑑𝑡, respectively, define the following limiting averages, when they exist:𝐻∶𝜆=∶l=im_𝑡→∞lim_𝑡→∞𝑡⁻¹𝑁(𝑡),𝑡⁻¹^𝑡0𝐺∶=lim_𝑛→∞𝐻(𝑠)𝑑𝑠,𝑛^𝑛−1_𝑘=1𝐺_𝑘.(2.1) Following Stidham Jr. [7], Heyman and Stidham Jr. [8] and El-Taha and Stidham Jr. [9] suppose that the bivariate sequence {(𝑇_𝑘,𝑓_𝑘(⋅)),𝑘≥1} satisfies the following condition.

Condition A. There exists a sequence {𝑊_𝑘,𝑘≥1} such that(i)𝑊_𝑘/𝑇_𝑘→0 as 𝑘→∞; and(ii)𝑓_𝑘(𝑡)=0 for 𝑡∉[𝑇_𝑘,𝑇_𝑘+𝑊_𝑘).

In economic terms, Condition A says that all the cost associated with the 𝑘th point (e.g., the 𝑘th customer) is incurred in a finite time interval beginning at the point (e.g., the arrival of the customer), and that the lengths of these intervals cannot grow at the same rate as the points themselves, as 𝑘→∞. This is a stronger-than-necessary condition for 𝐻=𝜆𝐺, but it is satisfied in most applications to queueing systems, in which the time points 𝑇_𝑘 and 𝑇_𝑘+𝑊_𝑘 correspond to customer arrivals and departures, respectively, and it is natural to assume that customers can only incur cost while they are physically present in the system. The following theorem is given by El-Taha and Stidham Jr. [9, Chapter 6].

Theorem 2.1. Suppose 𝑡⁻¹𝑁(𝑡)→𝜆 as 𝑡→∞, where 0≤𝜆≤∞, and Condition 𝐴 holds. Then if 𝑛⁻¹∑^𝑛𝑘=1𝐺_𝑘→𝐺 as 𝑛→∞, where 0≤𝐺≤∞, then 𝑡⁻¹∫^𝑡0𝐻(𝑠)𝑑𝑠→𝐻 as 𝑡→∞, and 𝐻=𝜆𝐺, provided 𝜆G is well defined.

We next consider two applications of 𝐻=𝜆𝐺. The first deals with residual service times, and the second application is the well-known Pollaczek-Khintchine formula for M/G/1 queues.

Residual Service Times
Let 𝑇_𝑘 with 𝑇₀=0, 𝑇_𝑘≤𝑇_𝑘+1, 𝑘=0,… be a deterministic point process, and 𝐴_𝑘=𝑇_𝑘+1−𝑇_𝑘 be the 𝑘th interevent. Let 𝑅(𝑠) be the residual time, that is, time until next event, that is, 𝑅(𝑠)=^∞_𝑘=1𝑇_𝑘+1−𝑠𝟏𝑇_𝑘≤𝑠<𝑇_𝑘+1.(2.2) Define the following limits when they exist: 𝐸𝐴=lim_𝑛→∞𝑛^𝑛−1_𝑘=1𝐸𝐴₂𝐴_𝑘;=lim_𝑛→∞𝑛^𝑛−1_𝑘=1𝑅=lim_𝑡→∞𝐴^2𝑘;𝑡⁻¹^𝑡0𝑅(𝑠)𝑑𝑠.(2.3) We interpret 𝐸𝐴 as the asymptotic average time between events, 𝐸𝐴² as the asymptotic second moment of the time between events, and 𝑅 as the asymptotic (long-run) time-average residual time of 𝑅(𝑠). In a queueing setting, it is sometimes useful to think of 𝐴_𝑘 as the service time of 𝑘th arrival, and 𝑅(𝑠) as the residual service time of the customer in service at time 𝑠, but we shall see other examples. Now, we state the following preliminary result.

Lemma 2.2. The asymptotic average residual time is 𝑅=𝐸𝐴²2𝐸𝐴.(2.4)

Proof. This proof uses 𝐻=𝜆𝐺. Let 𝑓_𝑘(𝑡)=(𝑇_𝑘+1−𝑡)𝟏{𝑇_𝑘≤𝑡<𝑇_𝑘+1}. Now, 𝐻(𝑡)=∑^∞𝑘=1𝑓_𝑘(𝑡) is the residual time at a randomly given time, and 𝐻 is the asymptotic residual time. Moreover, 𝐺_𝑘=∫^∞0𝑓_𝑘(𝑡)𝑑𝑡=𝐴^2𝑘/2; thus, 𝐺=lim_𝑛→∞𝑛⁻¹∑^𝑛𝑘=1𝐴^2𝑘/2=𝐸𝐴²/2. Therefore (with 𝜆=1/𝐸𝐴) 𝐻=𝜆𝐸𝐴²2=2𝐸𝐸𝐴𝐴²,(2.5) which completes the proof.

An alternative proof of this result uses the simpler relation, 𝑌=𝜆𝑋, of (El-Taha and Stidham Jr. [9]). Now, consider an M/G/1 queue with multiple vacations as defined in Section 3. Let 𝑆_𝑘 and 𝐷_𝑘 be the service requirement and queueing delay of 𝑘th arrival, respectively, and let 𝑓_𝑘(𝑡)=(𝑆_𝑘−(𝑡−𝑇_𝑘−𝐷_𝑘))𝟏{𝑇_𝑘+𝐷_𝑘≤𝑡<𝑇_𝑘+𝐷_𝑘+𝑆_𝑘}. Then we immediately see that the mean residual service time of the customer in service is given by 𝑅=𝜆𝐸𝑆²/2, noting that 𝜆=1/𝐸𝐴 in Lemma 2.2. Note that 𝑅 may also be written as (𝜌=𝜆𝐸𝑆)𝑅=𝜌𝐸𝑆²2𝐸𝑆(2.6) which is the product of the probability that the server is busy and the mean residual service time. Similarly, thinking of 𝑇_𝑘 as time instants when a server takes a vacation, the asymptotic mean residual vacation time, 𝑉_𝑅, is also given by 𝑉_𝑅=𝐸𝑉²2𝐸𝑉.(2.7)

Pollaczek-Khintchine Formula
Consider any G/G/1-FIFO queue. Let 𝑇_𝑘, 𝑘≥1 be the time instant of 𝑘th arrival; 𝑆_𝑘 and 𝐷_𝑘 be the 𝑘th arrival service requirement and delay (time in queue) respectively. Also let 𝑁(𝑡)=max{𝑘∶𝑇_𝑘≤𝑡} be the total arrivals during [0,𝑡]. Define the following limits when they exist: 𝜆=lim_𝑡→∞𝑁(𝑡)𝐸𝑆=lim_𝑛→∞𝑡,𝑛^𝑛−1_𝑘=1𝐸𝑆₂𝑆_𝑘,=lim_𝑛→∞𝑛^𝑛−1_𝑘=1𝐸𝐷=lim_𝑛→∞𝑆^2𝑘,𝑛^𝑛−1_𝑘=1𝐸𝐷₂𝐷_𝑘,=lim_𝑛→∞𝑛^𝑛−1_𝑘=1𝐸𝑆𝐷=lim_𝑛→∞𝐷^2𝑘,𝑛^𝑛−1_𝑘=1𝑆_𝑘𝐷_𝑘.(2.8) Note that we use the suggestive expectation notation even though the quantities are defined pathwise. Let 𝑓_𝑘(𝑡)=⎧⎪⎪⎨⎪⎪⎩𝑆^𝑘,if𝑇_𝑘≤𝑡<𝑇_𝑘+𝐷_𝑘𝑆_𝑘−𝑡−𝑇_𝑘−𝐷_𝑘,if𝑇_𝑘+𝐷_𝑘≤𝑡<𝑇_𝑘+𝐷_𝑘+𝑆_𝑘0,otherwise.(2.9) Now, 𝐻(𝑡)=^∞_𝑘=1𝑓_𝑘(𝑡)(2.10) is the total amount of work in the system at a randomly given time, and 𝐻 is the asymptotic average amount of work in the system. Moreover, 𝐺_𝑘=^∞0𝑓_𝑘(𝑡)𝑑𝑡=𝑆_𝑘𝐷_𝑘+𝑆^2𝑘2,(2.11) so that 𝐺=lim_𝑛→∞𝑛^𝑛−1_𝑘=1𝑆^𝑘𝐷_𝑘+𝑆^2𝑘2=𝐸𝑆𝐷+𝐸2𝑆².(2.12) Therefore, 𝐻=𝜆𝐸𝑆𝐷+𝐸𝑆²2,(2.13) The first term of the r.h.s. is the total amount of work in the system associated with customers waiting (excluding the one in service), and the second term is the residual service time of the customer in service.

M/G/1 Model
For an M/G/1-queue, we make the additional assumptions that 𝑆 and 𝐷 are independent, and arrivals are Poisson. We also assume that 𝜌=𝜆𝐸𝑆<1. Then by PASTA, we have 𝐸𝐷=𝐻 which implies 𝐸𝐷=𝜆(𝐸𝑆)(𝐸𝐷)+𝜆𝐸𝑆²/2. Therefore, 𝐸𝐷=𝜆𝐸𝑆²2(1−𝜌).(2.14)
In the next section, we extend Pollaczek-Khintchine formula to systems with multiple vacations using 𝐻=𝜆𝐺 and maintaining a pure sample path approach.

3. Systems with Multiple Vacations

Consider any G/G/1-FIFO stable queue with multiple vacations of length {𝑉_𝑘,𝑘≥1}. The server takes a vacation as soon as the number of customers in the system drops to 0 and continues to take successive vacations until the state 𝑁, where 𝑁 is the number of customers in the system, is ≥1 when the server returns from vacation. The process of taking vacations is repeated at the end of the following busy period. A vacation is not initiated when there are customers in the system. Let {𝑣_𝑘,𝑘=1,2,…} be the start time of 𝑘th vacation. Let 𝑀(𝑡) be the number of vacations up to time 𝑡 so that 𝑀(𝑡)=max{𝑘∶𝑣_𝑘≤𝑡}. Define the following limits when they exist: 𝛾=lim_𝑡→∞𝑀(𝑡)𝐸𝑉=lim_𝑛→∞𝑡,𝑛^𝑛−1_𝑘=1𝐸𝑉₂𝑉_𝑘,=lim_𝑛→∞𝑛^𝑛−1_𝑘=1𝑉^2𝑘.(3.1)

Let 𝜌∶=lim_𝑡→∞𝑡⁻¹∑^{𝑘𝑁(𝑡)}𝑆_𝑘, then the system is stable if  𝜌=𝜆𝐸𝑆<1. Note that the 𝜌 represents the long-run fraction of time the server is busy and 1−𝜌 is the long-run fraction of time the server is idle (i.e., on vacation).

Definition 3.1. The two sequences {𝑆_𝑘,𝑘≥1} and {𝐷_𝑘,𝑘≥1} are said to be asymptotically pathwise uncorrelated if 𝐸𝑆𝐷=𝐸𝑆𝐸𝐷.
Note that the requirement that𝐸𝑆𝐷=𝐸𝑆𝐸𝐷 is weaker than the corresponding stochastic assumption that the random variables 𝑆 and 𝐷, representing customer service times and delays, respectively, are independent.

Theorem 3.2. Consider the G/G/1-FIFO multivacation model. Suppose that 𝐸𝐷<∞ and 𝐸𝑉<∞, and that {𝑆_𝑘,𝑘≥1} and {𝐷_𝑘,𝑘≥1} are asymptotically pathwise uncorrelated. Then virtual delay in the system is given by 𝐻=𝜌𝐸𝐷+𝜌𝐸𝑆²2𝐸𝑆+(1−𝜌)2𝐸𝐸𝑉𝑉².(3.2)

Proof. Let 𝑇_𝑘 and 𝑣_𝑘 be time instants of customer arrivals, and vacation starts, respectively. Let 𝐴={𝑇_𝑘,𝑘=1,2,…}, 𝐵={𝑣_𝑘,𝑘=1,2,…}. Note that 𝐴∩𝐵=𝜙. Let 𝑓^1𝑘(𝑡) and 𝑓^2𝑘(𝑡) be defined such that 𝑓^1𝑘(𝑡)=𝑆_𝑘𝟏𝑇_𝑘≤𝑡<𝑓𝑇_k2𝑘+𝐷_𝑘+𝑆_𝑘−𝑡−𝑇_𝑘−𝐷_𝑘𝟏𝑇_𝑘+𝐷_𝑘≤𝑡<𝑇_𝑘+𝐷_𝑘+𝑆_𝑘;(𝑡)=𝑉_𝑘−𝑡−𝑣_𝑘𝟏𝑣_𝑘≤𝑡<𝑣_𝑘+𝑉_𝑘.(3.3) Note that 𝑓^1𝑘(𝑡) is the same as the 𝑓_𝑘(𝑡) given by (2.9). Here, 𝑓^1𝑘(𝑡) is the work remaining to be done for the 𝑘th arrival at time 𝑡; 𝑓^2𝑘(𝑡) is the vacation time remaining for the 𝑘th vacation at time 𝑡. Consider the bivariate sequence {𝑇_𝑘,𝑓^1𝑘(𝑡)} and let 𝐻¹(𝑡)∶=∑^∞𝑘=1𝑓^1𝑘(𝑡), so that 𝐻¹(𝑡) is the total amount of work in the system at time 𝑡. Using 𝐻=𝜆𝐺, we obtain (see (2.13)) 𝐻¹∶=lim_𝑡→∞𝑡⁻¹^𝑡0𝐻¹(𝑡)𝑑𝑡=𝜆𝐸𝑆𝐷+𝐸𝑆²2.(3.4) Similarly, consider the second bivariate sequence {𝑣_𝑘,𝑓^2𝑘(𝑡)} and let 𝐻²(𝑡)∶=∑^∞𝑘=1𝑓^2𝑘(𝑡), 𝐺^2𝑘=∫^∞0𝑓^2𝑘(𝑡)𝑑𝑡=𝑉^2𝑘/2. Here, 𝐻²(𝑡) is the residual vacation time in the system at time 𝑡, and 𝐺^2𝑘 is the remaining vacation time encountered by the 𝑘th arrival. Applying 𝐻=𝜆𝐺, we obtain 𝐺²=lim_𝑛→∞𝑛^𝑛−1𝐻₂_𝑘=1𝑉2^2𝑘=𝐸2𝑉²,∶=lim_𝑡→∞𝑡⁻¹^𝑡0𝐻²(𝑡)𝑑𝑡=𝛾𝐸𝑉²2,(3.5) where 𝛾 is the unconditional vacation rate. We need to compute 𝛾.
Recall that 𝑀(𝑡)=max{𝑘∶𝑣_𝑘≤𝑡} is the number of vacations up to time 𝑡. Let 𝐼(𝑡) be the status of the server at time 𝑡, that is, 𝐼(𝑡)=0 if the server is idle at time 𝑡 and 1 otherwise. Note that the server can be idle (on vacation) while customers are waiting for a vacation to end. It follows that 𝐸𝑉∶=lim_𝑛→∞𝑛^𝑛−1_𝑘=1𝑉_𝑘=lim_𝑡→∞𝑀(𝑡)^{−1𝑀(𝑡)}_𝑘=1𝑉_𝑘=lim_𝑡→∞𝑀(𝑡)⁻¹^𝑡0=lim_𝑡→∞𝟏{𝐼(𝑡)=0}𝑑𝑡𝑡𝑀(𝑡)⁻¹(1−𝜌),(3.6) so that 𝛾∶=lim_𝑡→∞𝑡⁻¹𝑀(𝑡)=1−𝜌𝐸𝑉.(3.7) Therefore, 𝐻²=(1−𝜌)𝐸𝑉²2𝐸𝑉.(3.8)
Let 𝐻(𝑡)∶=𝐻¹(𝑡)+𝐻²(𝑡) be the virtual delay, that is, total amount of work and remaining vacation time in the system at time 𝑡, and 𝐻=lim_𝑡→∞𝑡⁻¹∫^𝑡0𝐻(𝑠)𝑑𝑠 is the asymptotic (long-run) average amount of work and residual vacation in the system, in other words, 𝐻 is the asymptotic average virtual delay for a randomly arriving customer. Therefore, 𝐻∶=𝐻¹+𝐻²=𝜆𝐸𝑆𝐷+𝜆𝐸𝑆²2+(1−2𝐸𝜌)𝑉𝐸𝑉².(3.9) Now using the condition that 𝑆_𝑘 and 𝐷_𝑘 are asymptotically pathwise uncorrelated and simplifying, we obtain the result.

Equation (3.9) gives virtual delay 𝐻 under conditions weaker than those given in Theorem 3.2. Specifically, (3.9) holds without the assumption that 𝑆_𝑘 and 𝐷_𝑘 are asymptotically pathwise uncorrelated.

M/G/1 Queue with Vacations
Suppose that 𝑆 and 𝐷 are independent, arrivals are Poisson, and vacations 𝑉_𝑘 are 𝑖.𝑖.𝑑. having a general distribution function with mean 𝐸𝑉. Also assume that the system is stable, that is, 𝜌=𝜆𝐸𝑆<1. Then, by PASTA, we have 𝐸𝐷=𝐻, so Theorem 3.2 implies 𝐸𝐷=𝜌𝐸𝐷+𝜌𝐸𝑆²2𝐸𝑆+(1−2𝐸𝜌)𝑉𝐸𝑉².(3.10) Therefore, 𝐸𝐷=𝜆𝐸𝑆²2(1−𝜌)+2𝐸𝐸𝑉𝑉².(3.11) For systems with exponential vacations 𝐸𝐷=𝜆𝐸𝑆²2(1−𝜌)+𝐸𝑉.(3.12) Additionally, if service times are exponential with 𝐸𝑆=1/𝜇, we obtain 𝐸𝐷=𝜌𝜇(1−𝜌)+𝐸𝑉.(3.13)

4. Conservation Laws

Now we consider multiclass G/G/1 queue with multiple vacations, that is, a single-server system with general (not necessarily 𝑖.𝑖.𝑑.) interarrival and service times. Conservation laws hold for a wide variety of service disciplines like FIFO, LCFS, service in random order, and other priority rules. For nonvacation models, any scheduling rule has to be work-conserving, and the server is never idle when there is work in the system, but for vacation models this is not possible due to server vacation so the rule is adapted so that the server is never idle except when on vacation. We define a work-conserving scheduling rule similar to El-Taha and Stidham Jr. [9, pages 204–211], but adapted for a server that takes vacations.

Work-Conserving Scheduling Rules
Consider a bivariate sequence, {(𝑇_𝑛,𝑆_𝑛),𝑛=1,2,…}, where 𝑇_𝑛 and 𝑆_𝑛 are the arrival instant and work requirement, respectively, of 𝑛th arrival. A scheduling rule is said to be nonanticipative if the decision about which job to process at time 𝑡 depends only on {(𝑇_𝑛,𝑆_𝑛),𝑛=1,2,…,𝑁(𝑡)}, where 𝑁(𝑡)=max{𝑛∶𝑇_𝑛≤𝑡}, and possibly on decisions taken before time 𝑡. It is nonidling if the server does not take a vacation when there is at least one job in the system. Definition 4.1. A single-server model with nonpreemptive work-conserving scheduling (WCS) rules consists of(i)a single-server working at unit rate,(ii)a set of non-anticipative and non-idling scheduling rules.
We also assume that the scheduling rules are service time independent, and nonregenerative in the sense that the decision to schedule a job does not use any information from previous busy cycles. It is immediate from the definition of a work-conserving scheduling system that 𝑈(𝑡)=∑^∞𝑛=1𝑓^1𝑘(𝑡), the total work in the system at time 𝑡 is invariant with respect to WCS rules. It follows that the limiting average total work in the system, 𝑈=lim_𝑡→∞𝑡⁻¹^𝑡0𝑈(𝑠)𝑑𝑠,(4.1) is also invariant. The residual vacation at time 𝑡 is defined by 𝑉(𝑡)=∑^∞𝑘=1𝑓^2𝑘(𝑡), and the asymptotic (long-run) time-average residual vacation is given by 𝑉_𝑅=lim_𝑡→∞𝑡⁻¹^𝑡0𝑉(𝑠)𝑑𝑠.(4.2) So 𝑉_𝑅 is also invariant. Now let 𝐻 be the limiting average virtual delay in the system as defined in Theorem 3.2. Note that 𝐻 and 𝑈 are different due to the fact that the server takes multiple vacations. It is straight forward to see that 𝐻(𝑡)=𝑈(𝑡)+𝑉(𝑡), therefore, 𝐻=𝑈+𝑉_𝑅,(4.3) so that 𝐻 is invariant. Therefore, we have the following.Corollary 4.2. Consider the G/G/1 multivacation model with WCS rules that are nonpreemptive, regenerative and service time independent. Suppose that ED<∞ and 𝐸𝑉<∞, and that {𝑆_𝑘,𝑘≥1} and {𝐷_𝑘,𝑘≥1} are asymptotically pathwise uncorrelated. Then virtual delay in the system is given by 𝐻=𝜌𝐸𝐷+𝜌𝐸𝑆²2𝐸𝑆+(1−𝜌)2𝐸𝐸𝑉𝑉².(4.4)Basically, this Corollary extends Theorem 3.2 from FIFO to any nonpreemptive WCS rule. Next, we focus attention on multiclass systems.Multiclass Multiple Vacation Model
Consider the multi-class GI/GI/1 multiple vacation model as given in Section 3. Suppose that the discipline is work conserving when the server is not on vacation, when the server starts service, it continues until all work is cleared before taking vacation. The scheduling rule is nonanticipative and independent of previous arrival and service times, and within each class 𝑗, 𝑗=1,…,𝐽. That is, assume WCS rules. We give the following conservation law for a multi-class single-server system with multiple vacations.Theorem 4.3. 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 𝐸𝐷<∞ and 𝐸𝑉<∞, and that for each 𝑗{𝑆_𝑘𝑗,𝑘≥1} and {𝐷_𝑘𝑗,𝑘≥1} are asymptotically pathwise uncorrelated. The vector (𝐸𝐷₁,…,𝐸𝐷_𝐽) of expected actual queue delays per customer satisfies the following conservation law: ^𝐽_j=1𝜌_𝑗𝐸𝐷_𝑗=𝐻−^𝐽_𝑗=1𝜌_𝑗𝐸𝑆^2𝑗2𝐸𝑆_𝑗−(1−𝜌)𝐸𝑉²2𝐸𝑉.(4.5)Proof. Let 𝐻_𝑗 and 𝑈_𝑗, 𝑗=1,…,𝐽 be the class 𝑗 virtual delay and work in the system respectively, where 𝐻=∑^𝐽𝑗=1𝐻_𝑗 and 𝑈=∑^𝐽𝑗=1𝑈_𝑗. Now (4.3) becomes ^𝐽_𝑗=1𝐻_𝑗=^𝐽_𝑗=1𝑈_𝑗+𝑉_𝑅,(4.6) Recall that 𝐻, 𝑈, and 𝑉_𝑅 are invariant with respect to WCS rules. For each 𝑗, 𝑗=1,…,𝐽, let 𝜆_𝑗, 𝑆_𝑗, and 𝐷_𝑗 be the class 𝑗 mean arrival rate, service time, and queue delay, respectively. Also let 𝜌_𝑗=𝜆_𝑗𝐸𝑆_𝑗, and suppose that ∑^𝐽𝑗=1𝜌_𝑗<1. Then it follows from (3.4) 𝐸𝑈_𝑗=𝜆_𝑗𝐸𝑆^𝑗𝐷_𝑗+𝐸𝑆^2𝑗2.(4.7) Using the fact that the 𝑆_𝑗 and 𝐷_𝑗 are asymptotically pathwise uncorrelated and 𝜌_𝑖=𝜆_𝑗𝐸𝑆_𝑗, we have 𝐸𝑈_𝑗=𝜌_𝑗𝐸𝐷_𝑗+𝜌^𝑗𝐸𝑆^2𝑗2𝐸𝑆_𝑗.(4.8) Let 𝐵_𝑗=𝐸𝑆^2𝑗/2𝐸𝑆_𝑗. It follows by (4.6) and (4.8) that 𝐻=^𝐽_𝑗=1𝜌_𝑗𝐸𝐷_𝑗+^𝐽_𝑗=1𝜌_𝑗𝐵_𝑗+(1−𝜌)𝐸𝑉²2𝐸𝑉(4.9) is invariant over all WCS rules, since we have working conserving scheduling rules. Now let 𝜆=∑^𝐽𝑗=1𝜆_𝑗, and let 𝑆 be a random variable with distribution function 𝐹(𝑡)=∑^𝐽𝑗=1(𝜆_𝑗/𝜆)𝑃(𝑆_𝑗≤𝑡). Then ^𝐽_𝑗=1𝜌_𝑗𝐵_𝑗=𝜆𝐸𝑆²2(4.10) is also invariant over all WCS rules. Thus ∑^𝐽𝑗=1𝜌_𝑗𝐸𝐷_𝑗 is also invariant, and we have the following conservation law satisfied by the vector (𝐸𝐷₁,…,𝐸𝐷_𝐽) of expected queue delays ^𝐽_𝑗=1𝜌_𝑗𝐸𝐷_𝑗=𝐻−^𝐽_𝑗=1𝜌_𝑗𝐵_𝑗−(1−𝜌)𝐸𝑉²2𝐸𝑉(4.11) which leads to (4.5), thus proving the theorem.In the M/G/1 multi-class multiple vacation model, we derive an explicit expression for 𝐻=𝐸𝐷, using the fact that 𝐻 is invariant and, therefore, is equal to the delay in the M/G/1-FIFO discipline which is given by (3.11). Hence, the conservation equation (4.5) becomes, by substituting (3.11) and (4.10) in (4.5), ^𝐽_𝑗=1𝜌_𝑗𝐸𝐷_𝑗=𝜆𝐸𝑆²2(1−𝜌)+2𝐸𝐸𝑉𝑉²−𝜆𝐸2𝑆²−(1−2𝐸𝜌)𝑉𝐸𝑉²,(4.12) which simplifies to ^𝐽_𝑗=1𝜌_𝑗𝐸𝐷_𝑗=𝜆𝜌𝐸𝑆²2(1−𝜌)+𝜌2𝐸𝐸𝑉𝑉².(4.13) Theorem 4.3 can be used to construct conservation laws for waiting time in the system 𝑊_𝑗, number of customers in the system 𝐿_𝑗, and number of customers in the queue 𝐿^𝑞𝑗 using the fact that 𝐸𝑊_𝑗=𝐸𝐷_𝑗+𝐸𝑆_𝑗 and Little's formula.In this paper, we give two primary results. The first result presented in Theorem 3.2 gives a relation between virtual delay and actual delay for single-server multiple vacation model under conditions that are weaker than those given in the literature. This result is extended to multi-class models where a conservation law is given in Theorem 4.3 which is our second result. We use sample path analysis which allows us to give rigorous arguments by focusing on one realization of the stochastic process that describes the system evolution.