Research Article | Open Access
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.
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 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  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  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.  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  who presents a simple method for obtaining conservation laws for single-server queues with batch arrivals and multiple classes using the ASTA (El-Taha ) property. Relevant references also include Green and Stidham , Boxma and Groenendijk , and Choudhury . There is a vast literature on queues with vacations, and conservation laws. The reader may consult Takagi  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 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 which is a generalization of the well-known Little's formula.
We are given a deterministic sequence of time points, , with , , and we define , , so that is the number of points in . We assume that as , so that there are only a finite number of events in any finite time interval ( for all ), and we note that as , since for all . Associated with each time point , there is a function . The bivariate sequence constitutes the basic data, in terms of which the behavior of the system is described. We assume that is Lebesgue integrable on , for each .
With and defined by and , respectively, define the following limiting averages, when they exist: Following Stidham Jr. , Heyman and Stidham Jr.  and El-Taha and Stidham Jr.  suppose that the bivariate sequence satisfies the following condition.
Condition A. There exists a sequence such that(i) as ; and(ii) for .
In economic terms, Condition A says that all the cost associated with the th point (e.g., the th 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 . This is a stronger-than-necessary condition for , but it is satisfied in most applications to queueing systems, in which the time points and 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 as , where , and Condition holds. Then if as , where , then as , and , provided is well defined.
We next consider two applications of . 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 with , , be a deterministic point process, and be the th interevent. Let be the residual time, that is, time until next event, that is, Define the following limits when they exist: We interpret as the asymptotic average time between events, as the asymptotic second moment of the time between events, and as the asymptotic (long-run) time-average residual time of . In a queueing setting, it is sometimes useful to think of as the service time of th arrival, and as the residual service time of the customer in service at time , 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 . Let . Now, is the residual time at a randomly given time, and is the asymptotic residual time. Moreover, ; thus, . Therefore (with ) which completes the proof.
An alternative proof of this result uses the simpler relation, , of (El-Taha and Stidham Jr. ). Now, consider an M/G/1 queue with multiple vacations as defined in Section 3. Let and be the service requirement and queueing delay of th arrival, respectively, and let . Then we immediately see that the mean residual service time of the customer in service is given by , noting that in Lemma 2.2. Note that 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 as time instants when a server takes a vacation, the asymptotic mean residual vacation time, , is also given by
Consider any G/G/1-FIFO queue. Let , be the time instant of th arrival; and be the th arrival service requirement and delay (time in queue) respectively. Also let be the total arrivals during . Define the following limits when they exist: Note that we use the suggestive expectation notation even though the quantities are defined pathwise. Let Now, is the total amount of work in the system at a randomly given time, and is the asymptotic average amount of work in the system. Moreover, so that 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.
For an M/G/1-queue, we make the additional assumptions that and are independent, and arrivals are Poisson. We also assume that . Then by PASTA, we have which implies . Therefore,
In the next section, we extend Pollaczek-Khintchine formula to systems with multiple vacations using 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 . 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 , where is the number of customers in the system, is 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 be the start time of th vacation. Let be the number of vacations up to time so that . Define the following limits when they exist:
Let , then the system is stable if . Note that the represents the long-run fraction of time the server is busy and is the long-run fraction of time the server is idle (i.e., on vacation).
Definition 3.1. The two sequences and are said to be asymptotically pathwise uncorrelated if .
Note that the requirement that is weaker than the corresponding stochastic assumption that the random variables and , representing customer service times and delays, respectively, are independent.
Theorem 3.2. Consider the G/G/1-FIFO multivacation model. Suppose that and , and that and are asymptotically pathwise uncorrelated. Then virtual delay in the system is given by
Proof. Let and be time instants of customer arrivals, and vacation starts, respectively. Let , . Note that . Let and be defined such that
Note that is the same as the given by (2.9). Here, is the work remaining to be done for the th arrival at time ; is the vacation time remaining for the th vacation at time . Consider the bivariate sequence and let , so that is the total amount of work in the system at time . Using , we obtain (see (2.13))
Similarly, consider the second bivariate sequence and let , . Here, is the residual vacation time in the system at time , and is the remaining vacation time encountered by the th arrival. Applying , we obtain
where is the unconditional vacation rate. We need to compute .
Recall that is the number of vacations up to time . Let be the status of the server at time , that is, if the server is idle at time and 1 otherwise. Note that the server can be idle (on vacation) while customers are waiting for a vacation to end. It follows that so that Therefore,
Let be the virtual delay, that is, total amount of work and remaining vacation time in the system at time , and is the asymptotic (long-run) average amount of work and residual vacation in the system, in other words, is the asymptotic average virtual delay for a randomly arriving customer. Therefore, Now using the condition that and are asymptotically pathwise uncorrelated and simplifying, we obtain the result.
M/G/1 Queue with Vacations
Suppose that and are independent, arrivals are Poisson, and vacations are . having a general distribution function with mean . Also assume that the system is stable, that is, . Then, by PASTA, we have , so Theorem 3.2 implies Therefore, For systems with exponential vacations Additionally, if service times are exponential with , we obtain
4. Conservation Laws
Now we consider multiclass G/G/1 queue with multiple vacations, that is, a single-server system with general (not necessarily ) 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, , where and are the arrival instant and work requirement, respectively, of th arrival. A scheduling rule is said to be nonanticipative if the decision about which job to process at time depends only on , where , and possibly on decisions taken before time . 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 , the total work in the system at time 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 is defined by , and the asymptotic (long-run) time-average residual vacation is given by So is also invariant. Now let be the limiting average virtual delay in the system as defined in Theorem 3.2. Note that and are different due to the fact that the server takes multiple vacations. It is straight forward to see that , therefore, so that 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 and , and that and 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.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 , . 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 and , and that for each and are asymptotically pathwise uncorrelated. The vector of expected actual queue delays per customer satisfies the following conservation law: Proof. Let and , be the class virtual delay and work in the system respectively, where and . Now (4.3) becomes Recall that , , and are invariant with respect to WCS rules. For each , , let , , and be the class mean arrival rate, service time, and queue delay, respectively. Also let , and suppose that . Then it follows from (3.4) Using the fact that the and are asymptotically pathwise uncorrelated and , we have Let . It follows by (4.6) and (4.8) that is invariant over all WCS rules, since we have working conserving scheduling rules. Now let , and let be a random variable with distribution function . Then is also invariant over all WCS rules. Thus is also invariant, and we have the following conservation law satisfied by the vector of expected queue delays 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 , using the fact that 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), which simplifies to Theorem 4.3 can be used to construct conservation laws for waiting time in the system , number of customers in the system , and number of customers in the queue using the fact that 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.
- G. Choudhury, “A batch arrival queue with a vacation time under single vacation policy,” Computers & Operations Research, vol. 29, no. 14, pp. 1941–1955, 2002.
- O. J. Boxma and W. P. Groenendijk, “Pseudo-conservation laws in cyclic-service systems,” Journal of Applied Probability, vol. 24, no. 4, pp. 949–964, 1987.
- T. C. Green and S. Stidham, Jr., “Sample-path conservation laws, with applications to scheduling queues and fluid systems,” Queueing Systems, vol. 36, no. 1–3, pp. 175–199, 2000.
- C. Bisdikian, “A note on the conservation law for queues with batch arrivals,” IEEE Transactions on Communications, vol. 41, no. 6, pp. 832–835, 1993.
- M. El-Taha, PASTA and Related Results, Encyclopedia of Operations Research and Management Science, John Wiley & Sons, New York, NY, USA, 2010.
- H. Takagi, Queueing Analysis, Volume 1: Vacation and Priority Systems, Part 1, North-Holland Publishing, Amsterdam, The Netherlands, 1991.
- S. Stidham, Jr., “A last word on ,” Operations Research, vol. 22, no. 2, pp. 417–421, 1974.
- D. P. Heyman and S. Stidham, Jr., “The relation between customer and time averages in queues,” Operations Research, vol. 28, no. 4, pp. 983–994, 1980.
- M. El-Taha and S. Stidham, Jr., Sample-Path Analysis of Queueing Systems, International Series in Operations Research & Management Science, Kluwer Academic Publishing, Boston, Mass, USA, 1999.
Copyright © 2011 Muhammad El-Taha. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.