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

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.


Introduction
Consider a single-server queue with multiple vacations general arrival process and generalservice 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

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 wellknown 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 constitutes the basic data, in terms of which the behavior of the system is described.We assume that respectively, define the following limiting averages, when they exist:

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.
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 .
, 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 T 0 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, Define the following limits when they exist:

2.3
We interpret EA as the asymptotic average time between events, EA 2 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 Proof.This proof uses H λG. Let is the residual time at a randomly given time, and H is the asymptotic residual time.Moreover, 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 Then we immediately see that the mean residual service time of the customer in service is given by R λES 2 /2, noting that λ 1/EA in Lemma 2.2.Note that R may also be written as 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

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:

2.8
Note that we use the suggestive expectation notation even though the quantities are defined pathwise.Let

2.9
Now, 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, Therefore, 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 λES 2 /2.Therefore, ED λES 2 2 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.

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:

3.1
Let ρ : lim t → ∞ t −1 N t k 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 .Note that the requirement that ESD 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 Proof.Let T k and v k be time instants of customer arrivals, and vacation starts, respectively.

3.3
Note that f 1 k t is the same as the f k t given by 2.9 .Here, f 1 k t is the work remaining to be done for the kth arrival at time t; f 2 k t is the vacation time remaining for the kth vacation at time t.Consider the bivariate sequence {T k , f 1 k t } and let H 1 t : , so that H 1 t is the total amount of work in the system at time t.Using H λG, we obtain see 2.13 Similarly, consider the second bivariate sequence {v k , f 2 k t } and let H 2 t : Here, H 2 t is the residual vacation time in the system at time t, and G 2 k is the remaining vacation time encountered by the kth arrival.Applying H λG, we obtain 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

ISRN Applied Mathematics
Therefore, Let H t : H 1 t H 2 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 t 0 H 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, 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.

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.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 1 f 1 k 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, is also invariant.The residual vacation at time t is defined by V t ∞ k 1 f 2 k t , and the asymptotic long-run time-average residual vacation is given by So V R is also invariant.Now let H be the limiting average virtual delay in the system as defined in Theorem 3.

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 multiclass 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 ED 1 , . . ., ED J of expected actual queue delays per customer satisfies the following conservation law: Proof.Let H j and U j , j 1, . . ., J be the class j virtual delay and work in the system respectively, where H J j 1 H j and U J j 1 U j .Now 4.3 becomes 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 j ES j , and suppose that J j 1 ρ j < 1.Then it follows from 3.4 Using the fact that the S j and D j are asymptotically pathwise uncorrelated and ρ i λ j ES j , we have EU j ρ j ED j ρ j ES is also invariant over all WCS rules.Thus J j 1 ρ j ED j is also invariant, and we have the following conservation law satisfied by the vector ED 1 , . . ., ED J of expected queue delays 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 , 12 which simplifies to 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 q j 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.

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.
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 Basically, this Corollary extends Theorem 3.2 from FIFO to any nonpreemptive WCS rule.Next, we focus attention on multiclass systems.