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, {𝑇𝑘,𝑘≥1}, with 0≤𝑇𝑘≤𝑇𝑘+1<∞, 𝑘≥1, and we define 𝑁(𝑡)∶=max{𝑘≥1∶𝑇𝑘≤𝑡}, 𝑡≥0, so that 𝑁(𝑡) is the number of points in [0,𝑡]. We assume that ğ‘‡ğ‘˜â†’âˆž as ğ‘˜â†’âˆž, 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 ğ‘¡â†’âˆž, since 𝑇𝑘<∞ for all 𝑘≥1. Associated with each time point 𝑇𝑘, there is a function 𝑓𝑘∶[0,∞)→[0,∞). The bivariate sequence {(𝑇𝑘,𝑓𝑘(⋅)),𝑘≥1} constitutes the basic data, in terms of which the behavior of the system is described. We assume that 𝑓𝑘(𝑡) is Lebesgue integrable on 𝑡∈[0,∞), for each 𝑘≥1.

With 𝐻(𝑡) and 𝐺𝑘 defined by ∑𝐻(𝑡)=âˆžğ‘˜=1𝑓𝑘(𝑡) and 𝐺𝑘=∫∞0𝑓𝑘(𝑡)𝑑𝑡, respectively, define the following limiting averages, when they exist:𝜆∶=limğ‘¡â†’âˆžğ‘¡âˆ’1𝑁(𝑡),𝐻∶=limğ‘¡â†’âˆžğ‘¡âˆ’1𝑡0𝐻(𝑠)𝑑𝑠,𝐺∶=limğ‘›â†’âˆžğ‘›ğ‘›âˆ’1𝑘=1𝐺𝑘.(2.1) Following Stidham Jr. [7], Heyman and Stidham Jr. [8] and El-Taha and Stidham Jr. [9] suppose that the bivariate sequence {(𝑇𝑘,𝑓𝑘(⋅)),𝑘≥1} satisfies the following condition.

Condition A. There exists a sequence {𝑊𝑘,𝑘≥1} such that(i)𝑊𝑘/𝑇𝑘→0 as ğ‘˜â†’âˆž; and(ii)𝑓𝑘(𝑡)=0 for 𝑡∉[𝑇𝑘,𝑇𝑘+𝑊𝑘).

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 ğ‘˜â†’âˆž. 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 𝑇𝑘 and 𝑇𝑘+𝑊𝑘 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 ğ‘¡â†’âˆž, where 0â‰¤ğœ†â‰¤âˆž, and Condition 𝐴 holds. Then if 𝑛−1∑𝑛𝑘=1𝐺𝑘→𝐺 as ğ‘›â†’âˆž, where 0â‰¤ğºâ‰¤âˆž, then 𝑡−1∫𝑡0𝐻(𝑠)𝑑𝑠→𝐻 as ğ‘¡â†’âˆž, 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 𝑇𝑘 with 𝑇0=0, 𝑇𝑘≤𝑇𝑘+1, 𝑘=0,… be a deterministic point process, and 𝐴𝑘=𝑇𝑘+1−𝑇𝑘 be the 𝑘th interevent. Let 𝑅(𝑠) be the residual time, that is, time until next event, that is, 𝑅(𝑠)=âˆžî“ğ‘˜=1𝑇𝑘+1𝟏𝑇−𝑠𝑘≤𝑠<𝑇𝑘+1.(2.2) Define the following limits when they exist: 𝐸𝐴=limğ‘›â†’âˆžğ‘›ğ‘›âˆ’1𝑘=1𝐴𝑘;𝐸𝐴2=limğ‘›â†’âˆžğ‘›ğ‘›âˆ’1𝑘=1𝐴2𝑘;𝑅=limğ‘¡â†’âˆžğ‘¡âˆ’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 𝐴𝑘 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=𝐸𝐴2.2𝐸𝐴(2.4)

Proof. This proof uses 𝐻=𝜆𝐺. Let 𝑓𝑘(𝑡)=(𝑇𝑘+1−𝑡)𝟏{𝑇𝑘≤𝑡<𝑇𝑘+1}. Now, ∑𝐻(𝑡)=âˆžğ‘˜=1𝑓𝑘(𝑡) is the residual time at a randomly given time, and 𝐻 is the asymptotic residual time. Moreover, 𝐺𝑘=∫∞0𝑓𝑘(𝑡)𝑑𝑡=𝐴2𝑘/2; thus, 𝐺=limğ‘›â†’âˆžğ‘›âˆ’1∑𝑛𝑘=1𝐴2𝑘/2=𝐸𝐴2/2. Therefore (with 𝜆=1/𝐸𝐴) 𝐻=𝜆𝐸𝐴22=𝐸𝐴2,2𝐸𝐴(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 𝑆𝑘 and 𝐷𝑘 be the service requirement and queueing delay of 𝑘th arrival, respectively, and let 𝑓𝑘(𝑡)=(𝑆𝑘−(𝑡−𝑇𝑘−𝐷𝑘))𝟏{𝑇𝑘+𝐷𝑘≤𝑡<𝑇𝑘+𝐷𝑘+𝑆𝑘}. 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 𝑇𝑘 as time instants when a server takes a vacation, the asymptotic mean residual vacation time, 𝑉𝑅, is also given by 𝑉𝑅=𝐸𝑉22𝐸𝑉.(2.7)

Pollaczek-Khintchine Formula
Consider any G/G/1-FIFO queue. Let 𝑇𝑘, 𝑘≥1 be the time instant of 𝑘th arrival; 𝑆𝑘 and 𝐷𝑘 be the 𝑘th arrival service requirement and delay (time in queue) respectively. Also let 𝑁(𝑡)=max{𝑘∶𝑇𝑘≤𝑡} be the total arrivals during [0,𝑡]. Define the following limits when they exist: 𝜆=limğ‘¡â†’âˆžğ‘(𝑡)𝑡,𝐸𝑆=limğ‘›â†’âˆžğ‘›ğ‘›âˆ’1𝑘=1𝑆𝑘,𝐸𝑆2=limğ‘›â†’âˆžğ‘›ğ‘›âˆ’1𝑘=1𝑆2𝑘,𝐸𝐷=limğ‘›â†’âˆžğ‘›ğ‘›âˆ’1𝑘=1𝐷𝑘,𝐸𝐷2=limğ‘›â†’âˆžğ‘›ğ‘›âˆ’1𝑘=1𝐷2𝑘,𝐸𝑆𝐷=limğ‘›â†’âˆžğ‘›ğ‘›âˆ’1𝑘=1𝑆𝑘𝐷𝑘.(2.8) Note that we use the suggestive expectation notation even though the quantities are defined pathwise. Let ğ‘“ğ‘˜âŽ§âŽªâŽªâŽ¨âŽªâŽªâŽ©ğ‘†(𝑡)=𝑘,if𝑇𝑘≤𝑡<𝑇𝑘+𝐷𝑘𝑆𝑘−𝑡−𝑇𝑘−𝐷𝑘,if𝑇𝑘+𝐷𝑘≤𝑡<𝑇𝑘+𝐷𝑘+𝑆𝑘0,otherwise.(2.9) Now, 𝐻(𝑡)=âˆžî“ğ‘˜=1𝑓𝑘(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, 𝐺𝑘=∞0𝑓𝑘(𝑡)𝑑𝑡=𝑆𝑘𝐷𝑘+𝑆2𝑘2,(2.11) so that 𝐺=limğ‘›â†’âˆžğ‘›ğ‘›âˆ’1𝑘=1𝑆𝑘𝐷𝑘+𝑆2𝑘2=𝐸𝑆𝐷+𝐸𝑆22.(2.12) Therefore, 𝐻=𝜆𝐸𝑆𝐷+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 {𝑉𝑘,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 {𝑣𝑘,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∶𝑣𝑘≤𝑡}. Define the following limits when they exist: 𝛾=limğ‘¡â†’âˆžğ‘€(𝑡)𝑡,𝐸𝑉=limğ‘›â†’âˆžğ‘›ğ‘›âˆ’1𝑘=1𝑉𝑘,𝐸𝑉2=limğ‘›â†’âˆžğ‘›ğ‘›âˆ’1𝑘=1𝑉2𝑘.(3.1)

Let 𝜌∶=limğ‘¡â†’âˆžğ‘¡âˆ’1∑𝑘𝑁(𝑡)𝑆𝑘, 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 {𝑆𝑘,𝑘≥1} and {𝐷𝑘,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 {𝑆𝑘,𝑘≥1} and {𝐷𝑘,𝑘≥1} are asymptotically pathwise uncorrelated. Then virtual delay in the system is given by 𝐻=𝜌𝐸𝐷+𝜌𝐸𝑆22𝐸𝑆+(1−𝜌)𝐸𝑉22𝐸𝑉.(3.2)

Proof. Let 𝑇𝑘 and 𝑣𝑘 be time instants of customer arrivals, and vacation starts, respectively. Let 𝐴={𝑇𝑘,𝑘=1,2,…}, 𝐵={𝑣𝑘,𝑘=1,2,…}. Note that 𝐴∩𝐵=𝜙. Let 𝑓1𝑘(𝑡) and 𝑓2𝑘(𝑡) be defined such that 𝑓1𝑘(𝑡)=𝑆𝑘𝟏𝑇𝑘≤𝑡<𝑇k+𝐷𝑘+𝑆𝑘−𝑡−𝑇𝑘−𝐷𝑘𝟏𝑇𝑘+𝐷𝑘≤𝑡<𝑇𝑘+𝐷𝑘+𝑆𝑘;𝑓2𝑘𝑉(𝑡)=𝑘−𝑡−𝑣𝑘𝟏𝑣𝑘≤𝑡<𝑣𝑘+𝑉𝑘.(3.3) Note that 𝑓1𝑘(𝑡) is the same as the 𝑓𝑘(𝑡) given by (2.9). Here, 𝑓1𝑘(𝑡) is the work remaining to be done for the 𝑘th arrival at time 𝑡; 𝑓2𝑘(𝑡) is the vacation time remaining for the 𝑘th vacation at time 𝑡. Consider the bivariate sequence {𝑇𝑘,𝑓1𝑘(𝑡)} and let 𝐻1∑(𝑡)∶=âˆžğ‘˜=1𝑓1𝑘(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∶=limğ‘¡â†’âˆžğ‘¡âˆ’1𝑡0𝐻1(𝑡)𝑑𝑡=𝜆𝐸𝑆𝐷+𝐸𝑆22.(3.4) Similarly, consider the second bivariate sequence {𝑣𝑘,𝑓2𝑘(𝑡)} and let 𝐻2∑(𝑡)∶=âˆžğ‘˜=1𝑓2𝑘(𝑡), 𝐺2𝑘=∫∞0𝑓2𝑘(𝑡)𝑑𝑡=𝑉2𝑘/2. Here, 𝐻2(𝑡) is the residual vacation time in the system at time 𝑡, and 𝐺2𝑘 is the remaining vacation time encountered by the 𝑘th arrival. Applying 𝐻=𝜆𝐺, we obtain 𝐺2=limğ‘›â†’âˆžğ‘›ğ‘›âˆ’1𝑘=1𝑉2𝑘2=𝐸𝑉22,𝐻2∶=limğ‘¡â†’âˆžğ‘¡âˆ’1𝑡0𝐻2(𝑡)𝑑𝑡=𝛾𝐸𝑉22,(3.5) where 𝛾 is the unconditional vacation rate. We need to compute 𝛾.
Recall that 𝑀(𝑡)=max{𝑘∶𝑣𝑘≤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∶=limğ‘›â†’âˆžğ‘›ğ‘›âˆ’1𝑘=1𝑉𝑘=limğ‘¡â†’âˆžğ‘€(𝑡)−1𝑀(𝑡)𝑘=1𝑉𝑘=limğ‘¡â†’âˆžğ‘€(𝑡)−1𝑡0𝟏{𝐼(𝑡)=0}𝑑𝑡=limğ‘¡â†’âˆžğ‘¡ğ‘€(𝑡)−1(1−𝜌),(3.6) so that 𝛾∶=limğ‘¡â†’âˆžğ‘¡âˆ’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=limğ‘¡â†’âˆžğ‘¡âˆ’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−𝜌)𝐸𝑉22𝐸𝑉.(3.9) Now using the condition that 𝑆𝑘 and 𝐷𝑘 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 𝑆𝑘 and 𝐷𝑘 are asymptotically pathwise uncorrelated.

M/G/1 Queue with Vacations
Suppose that 𝑆 and 𝐷 are independent, arrivals are Poisson, and vacations 𝑉𝑘 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+𝜌𝐸𝑆2+2𝐸𝑆(1−𝜌)𝐸𝑉22𝐸𝑉.(3.10) Therefore, 𝐸𝐷=𝜆𝐸𝑆2+2(1−𝜌)𝐸𝑉22𝐸𝑉.(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 𝜌𝐸𝐷=𝜇(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, {(𝑇𝑛,𝑆𝑛),𝑛=1,2,…}, where 𝑇𝑛 and 𝑆𝑛 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 {(𝑇𝑛,𝑆𝑛),𝑛=1,2,…,𝑁(𝑡)}, where 𝑁(𝑡)=max{𝑛∶𝑇𝑛≤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)=âˆžğ‘›=1𝑓1𝑘(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=limğ‘¡â†’âˆžğ‘¡âˆ’1𝑡0𝑈(𝑠)𝑑𝑠,(4.1) is also invariant. The residual vacation at time 𝑡 is defined by ∑𝑉(𝑡)=âˆžğ‘˜=1𝑓2𝑘(𝑡), and the asymptotic (long-run) time-average residual vacation is given by 𝑉𝑅=limğ‘¡â†’âˆžğ‘¡âˆ’1𝑡0𝑉(𝑠)𝑑𝑠.(4.2) So 𝑉𝑅 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+𝑉𝑅,(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 {𝑆𝑘,𝑘≥1} and {𝐷𝑘,𝑘≥1} are asymptotically pathwise uncorrelated. Then virtual delay in the system is given by 𝐻=𝜌𝐸𝐷+𝜌𝐸𝑆22𝐸𝑆+(1−𝜌)𝐸𝑉22𝐸𝑉.(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{𝑆𝑘𝑗,𝑘≥1} and {𝐷𝑘𝑗,𝑘≥1} are asymptotically pathwise uncorrelated. The vector (𝐸𝐷1,…,𝐸𝐷𝐽) of expected actual queue delays per customer satisfies the following conservation law: 𝐽j=1𝜌𝑗𝐸𝐷𝑗=𝐻−𝐽𝑗=1𝜌𝑗𝐸𝑆2𝑗2𝐸𝑆𝑗−(1−𝜌)𝐸𝑉22𝐸𝑉.(4.5)Proof. Let 𝐻𝑗 and 𝑈𝑗, 𝑗=1,…,𝐽 be the class 𝑗 virtual delay and work in the system respectively, where ∑𝐻=𝐽𝑗=1𝐻𝑗 and ∑𝑈=𝐽𝑗=1𝑈𝑗. Now (4.3) becomes 𝐽𝑗=1𝐻𝑗=𝐽𝑗=1𝑈𝑗+𝑉𝑅,(4.6) Recall that 𝐻, 𝑈, and 𝑉𝑅 are invariant with respect to WCS rules. For each 𝑗, 𝑗=1,…,𝐽, let 𝜆𝑗, 𝑆𝑗, and 𝐷𝑗 be the class 𝑗 mean arrival rate, service time, and queue delay, respectively. Also let 𝜌𝑗=𝜆𝑗𝐸𝑆𝑗, and suppose that ∑𝐽𝑗=1𝜌𝑗<1. Then it follows from (3.4) 𝐸𝑈𝑗=𝜆𝑗𝐸𝑆𝑗𝐷𝑗+𝐸𝑆2𝑗2.(4.7) Using the fact that the 𝑆𝑗 and 𝐷𝑗 are asymptotically pathwise uncorrelated and 𝜌𝑖=𝜆𝑗𝐸𝑆𝑗, we have 𝐸𝑈𝑗=𝜌𝑗𝐸𝐷𝑗+𝜌𝑗𝐸𝑆2𝑗2𝐸𝑆𝑗.(4.8) Let 𝐵𝑗=𝐸𝑆2𝑗/2𝐸𝑆𝑗. It follows by (4.6) and (4.8) that 𝐻=𝐽𝑗=1𝜌𝑗𝐸𝐷𝑗+𝐽𝑗=1𝜌𝑗𝐵𝑗+(1−𝜌)𝐸𝑉22𝐸𝑉(4.9) is invariant over all WCS rules, since we have working conserving scheduling rules. Now let ∑𝜆=𝐽𝑗=1𝜆𝑗, and let 𝑆 be a random variable with distribution function ∑𝐹(𝑡)=𝐽𝑗=1(𝜆𝑗/𝜆)𝑃(𝑆𝑗≤𝑡). Then 𝐽𝑗=1𝜌𝑗𝐵𝑗=𝜆𝐸𝑆22(4.10) is also invariant over all WCS rules. Thus ∑𝐽𝑗=1𝜌𝑗𝐸𝐷𝑗 is also invariant, and we have the following conservation law satisfied by the vector (𝐸𝐷1,…,𝐸𝐷𝐽) of expected queue delays 𝐽𝑗=1𝜌𝑗𝐸𝐷𝑗=𝐻−𝐽𝑗=1𝜌𝑗𝐵𝑗−(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=1𝜌𝑗𝐸𝐷𝑗=𝜆𝐸𝑆2+2(1−𝜌)𝐸𝑉2−2𝐸𝑉𝜆𝐸𝑆22−(1−𝜌)𝐸𝑉22𝐸𝑉,(4.12) which simplifies to 𝐽𝑗=1𝜌𝑗𝐸𝐷𝑗=𝜆𝜌𝐸𝑆2+2(1−𝜌)𝜌𝐸𝑉22𝐸𝑉.(4.13) Theorem 4.3 can be used to construct conservation laws for waiting time in the system 𝑊𝑗, number of customers in the system 𝐿𝑗, and number of customers in the queue ğ¿ğ‘žğ‘— using the fact that 𝐸𝑊𝑗=𝐸𝐷𝑗+𝐸𝑆𝑗 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.