An M/G/1 queue with cyclic service times

In this paper we consider a single server queue with Poisson arrivals and general service distributions in which the service distributions are changed cyclically according to customer sequence number. This model extends a previous study that used cyclic exponential service times to the treatment of general service distributions. First, the stationary probability generating function and the average number of customers in the system are found. Then, a single vacation queueing system with aN-limited service policy, in which the server goes on vacation after servingN consecutive customers is analyzed as a particular case of our model. Also, to increase the flexibility of using theM/G/1 model with cyclic service times in optimization problems, an approximation approach is introduced in order to obtain the average number of customers in the system. Finally, using this approximation, the optimalN-limited service policy for a single vacation queueing system is obtained.On leave from the Department of Industrial Engineering, Iran University of Science and Technology, Narmak, Tehran 16844, Iran.     

The MAP/(PH/PH)/1 queue with self-generation of priorities and non-preemptive service

Customers arriving according to a Markovian arrival process are served at a single server facility. Waiting customers generate priority at a constant rate γ$\gamma$; such a customer waits in a waiting space of capacity 1 if this waiting space is not already occupied by a priority generated customer; else it leaves the system. A customer in service will be completely served before the priority generated customer is taken for service (non-preemptive service discipline). Only one priority generated customer can wait at a time and a customer generating into priority at that time will have to leave the system in search of emergency service elsewhere. The service times of ordinary and priority generated customers follow PH-distributions. The matrix analytic method is used to compute the steady state distribution. Performance measures such as the probability of n consecutive services of priority generated customers, the probability of the same for ordinary customers, and the mean waiting time of a tagged customer are found by approximating them by their corresponding values in a truncated system. All these results are supported numerically.     

M/G/1/N vacation model with varying E-limited service discipline

We define and analyze anM/G/1/N vacation model that uses a service discipline that we call theE-limited with limit variation discipline. According to this discipline, the server provides service until either the system is emptied (i.e. exhausted) or a randomly chosen limit ofl customers has been served. The server then goes on a vacation before returning to service the queue again. The queue length distribution and the Laplace-Stieltjes transforms of the waiting time, busy period and cycle time distributions are found. Further, an expression for the mean waiting time is developed. Several previously analyzed service disciplines, including Bernoulli scheduling, nonexhaustive service and limited service, are special cases of the general varying limit discipline that is analyzed in this paper.     

Stationary distribution of queue length in G / M / 1 queue with two-stage service policy

We consider a G / M / 1 queue with two-stage service policy. The server starts to serve with rate of μ₁ customers per unit time until the number of customers in the system reaches λ. At this moment, the service rate is changed to that of μ₂ customers per unit time and this rate continues until the system is empty. We obtain the stationary distribution of the number of customers in the system.     

Sojourn and waiting times in a single-server system with state-dependent mean service rate

A birth-death queueing system with asingle server, first-come first-served discipline, Poisson arrivals and state-dependent mean service rate is considered. The problem of determining the equilibrium densities of the sojourn and waiting times is formulated, in general. The particular case in which the mean service rate has one of two values, depending on whether or not the number of customers in the system exceeds a prescribed threshold, is then investigated. A generating function is derived for the Laplace transforms of the densities of the sojourn and waiting times, leading to explicit expressions for these quantities. Explicit expressions for the second moments of the sojourn and waiting times are also obtained.     

AN M/G/1 RETRIAL QUEUE WITH SECOND MULTI-OPTIONAL SERVICE, FEEDBACK AND UNRELIABLE SERVER

An M/G/1 retrial queue with two-phase service and feedback is studied in this paper, where the server is subject to starting failures and breakdowns during service. Primary customers get in the system according to a Poisson process, and they will receive service immediately if the server is available upon arrival. Otherwise, they will enter a retrial orbit and are queued in the orbit in accordance with a first-come-first-served （FCFS） discipline. Customers are allowed to balk and renege at particular times. All customers demand the first “essential” service, whereas only some of them demand the second “multi-optional” service. It is assumed that the retrial time, service time and repair time of the server are all arbitrarily distributed. The necessary and sufficient condition for the system stability is derived. Using a supplementary variable method, the steady-state solutions for some queueing and reliability measures of the system are obtained.     

A two-queue model with Bernoulli service schedule and switching times

In this paper, we present a detailed analysis of a cyclic-service queueing system consisting of two parallel queues, and a single server. The server serves the two queues with a Bernoulli service schedule described as follows. At the beginning of each visit to a queue, the server always serves a customer. At each epoch of service completion in the ith queue at which the queue is not empty, the server makes a random decision: with probability p_i, it serves the next customer; with probability 1-p_i, it switches to the other queue. The server takes switching times in its transition from one queue to the other. We derive the generating functions of the joint stationary queue-length distribution at service completion instants, by using the approach of the boundary value problem for complex variables. We also determine the Laplace-Stieltjes transforms of waiting time distributions for both queues, and obtain their mean waiting times. This revised version was published online in June 2006 with corrections to the Cover Date.     

Scheduling service in tandem queues attended by a single server

Consider a tandem queue model with a single server who can switch instantaneously from one queue to another. Customers arrive according to a Poisson process with rate λ . The amount of service required by each customer at the i^th queue is an exponentially distributed random variable with rate μ_i. Whenever two or more customers are in the system, the decision as to which customer should be served first depends on the optimzation criterion. In this system all server allocation policies in the finite set of work conserving deterministic policies have the same expected first passage times (makespan) to empty the system of customers from any initial state. However, a unique policy maximizes the first passage probability of empty-ing the system before the number of customers exceeds K, for any value of K, and it stochastically minimizes (he number of customers in the system at any time t > 0 . This policy always assigns the server to the non empty queue closest to the exit     

Two queues with alternating service and server breakdown

We consider a queueing system with two stations served by a single server in a cyclic manner. We assume that at most one customer can be served at a station when the server arrives at the station. The system is subject to service interuption that arises from server breakdown. When a server breakdown occurs, the server must be repaired before service can resume. We obtain the approximate mean delay of customers in the system.     

Performance of service policies in a specialized service system with parallel servers

This paper considers a scheduling problem occurring in a specialized service system with parallel servers. In the system, customers are divided into the “ordinary” and “special” categories according to their service needs. Ordinary customers can be served by any server, while special customers can be served only by the flexible servers. We assume that the service time for any ordinary customer is the same and all special customers have another common service time. We analyze three classes of service policies used in practice, namely, policies with priority, policies without priority and mixed policies. The worst-case performance ratios are obtained for all of these service policies.     

MAP/(PH/PH)/c Queue with Self-Generation of Priorities and Non-Preemptive Service

Abstract

Customers arriving according to a Markovian arrival process are served at a c server facility. Waiting customers generate into priority while waiting in the system (self-generation of priorities), at a constant rate γ; such a customer is immediately taken for service, if at least one of the servers is free. Else it waits at a waiting space of capacity c exclusively for priority generated customers, provided there is vacancy. A customer in service is not preempted to accommodate a priority generated customer. The service times of ordinary and priority generated customers follow distinct PH-distributions. It is proved that the system is always stable. We provide a numerical procedure to compute the optimal number of servers to be employed to minimize the loss to the system. Several performance measures are evaluated.     

Priority queues with semi-exhaustive service

This paper studies a new type of multi-class priority queues with semi-exhaustive service and server vacations, which operates as follows: A single server continues serving messages in queuen until the number of messages decreases toone less than that found upon the server's last arrival at queuen, where 1nN. In succession, messages of the highest class present in the system, if any, will be served according to this semi-exhaustive service. Applying the delay cycle analysis and introducing a super-message composed of messages served in a busy period, we derive explicitly the Laplace-Stieltjes transforms (LSTs) and the first two moments of the message waiting time distributions for each class in the M/G/1-type priority queues with multiple and single vacations. We also derive a conversion relationship between the LSTs for waiting times in the multiple and single vacation models.     

Analysis of the M [X]/G/1 queues with second multi-optional service and unreliable server

A bulk-arrival single server queueing system with second multi-optional service and unreliable server is studied in this paper. Customers arrive in batches according to a homogeneous Poisson process, all customers demand the first ＂essential＂ service, whereas only some of them demand the second ＂multi-optional＂ service. The first service time and the second service all have general distribution and they are independent. We assume that the server has a service-phase dependent, exponentially distributed life time as well as a servicephase dependent, generally distributed repair time. Using a supplementary variable method, we obtain the transient and the steady-state solutions for both queueing and reliability measures of interest.     

Sojourn time analysis for a cyclic-service tandem queueing model with general decrementing service

A sojourn time analysis is provided for a cyclic-service tandem queue with general decrementing service which operates as follows: starting once a service of queue 1 in the first stage, a single server continues serving messages in queue 1 until either queue 1 becomes empty, or the number of messages decreases to k less than that found upon the server's last arrival at queue 1, whichever occurs first, where 1 ≤ k ≤ ∞. After service completion in queue 1, the server switches over to queue 2 in the second stage and serves all messages in queue 2 until it becomes empty. It is assumed that an arrival stream is Poissonian, message service times at each stage are generally distributed and switch-over times are zero. This paper analyzes joint queue-length distributions and message sojourn time distributions.     

An <Emphasis Type="Italic">M</Emphasis><Superscript>[<Emphasis Type="Italic">X</Emphasis>]</Superscript>/<Emphasis Type="Italic">G</Emphasis>/1 retrial queue with server breakdowns and constant rate of repeated attempts

We consider an M ^[X]/G/1 retrial queue subject to breakdowns where the retrial time is exponential and independent of the number of customers applying for service. If a coming batch of customers finds the server idle, one of the arriving customers begins his service immediately and the rest joins a retrial group (called orbit) to repeat his request later; otherwise, if the server is busy or down, all customers of the coming batch enter the orbit. It is assumed that the server has a constant failure rate and arbitrary repair time distribution. We study the ergodicity of the embedded Markov chain, its stationary distribution and the joint distribution of the server state and the orbit size in steady-state. The orbit and system size distributions are obtained as well as some performance measures of the system. The stochastic decomposition property and the asymptotic behavior under high rate of retrials are discussed. We also analyse some reliability problems, the k-busy period and the ordinary busy period of our retrial queue. Besides, we give a recursive scheme to compute the distribution of the number of served customers during the k-busy period and the ordinary busy period. The effects of several parameters on the system are analysed numerically. I. Atencia’s and Moreno’s research is supported by the MEC through the project MTM2005-01248.     

Analysis of a single server queue with semi-Markovian service interruption

In this paper we analyze a discrete-time single server queue where the service time equals one slot. The numbers of arrivals in each slot are assumed to be independent and identically distributed random variables. The service process is interrupted by a semi-Markov process, namely in certain states the server is available for service while the server is not available in other states. We analyze both the transient and steady-state models. We study the generating function of the joint probability of queue length, the state and the residual sojourn time of the semi-Markov process. We derive a system of Hilbert boundary value problems for the generating functions. The system of Hilbert boundary value problems is converted to a system of Fredholm integral equations. We show that the system of Fredholm integral equations has a unique solution. This revised version was published online in June 2006 with corrections to the Cover Date.     

A heavy-traffic analysis of a closed queueing system with a GI/∞ service center

This paper studies the heavy-traffic behavior of a closed system consisting of two service stations. The first station is an infinite server and the second is a single server whose service rate depends on the size of the queue at the station. We consider the regime when both the number of customers, n, and the service rate at the single-server station go to infinity while the service rate at the infinite-server station is held fixed. We show that, as n→∞, the process of the number of customers at the infinite-server station normalized by n converges in probability to a deterministic function satisfying a Volterra integral equation. The deviations of the normalized queue from its deterministic limit multiplied by √n converge in distribution to the solution of a stochastic Volterra equation. The proof uses a new approach to studying infinite-server queues in heavy traffic whose main novelty is to express the number of customers at the infinite server as a time-space integral with respect to a time-changed sequential empirical process. This gives a new insight into the structure of the limit processes and makes the end results easy to interpret. Also the approach allows us to give a version of the classical heavy-traffic limit theorem for the G/GI/∞ queue which, in particular, reconciles the limits obtained earlier by Iglehart and Borovkov. This revised version was published online in June 2006 with corrections to the Cover Date.     

The throughput performance of a prioritized LIFO service discipline

The LIFO service discipline maximizes throughput for certain queuing systems with delay-dependent customer behavior. To study the effects of priorities in such a system, we obtain delay distributions for each customer class of a K-class, single server system with nonpreemptive priorities and LIFO within each queue.     

Asymptotics of stochastic networks with subexponential service times

We analyse the tail behaviour of stationary response times in the class of open stochastic networks with renewal input admitting a representation as (max,+)-linear systems. For a K-station tandem network of single server queues with infinite buffer capacity, which is one of the simplest models in this class, we first show that if the tail of the service time distribution of one server, say server i ₀ ∈ {1,...,K}, is subexponential and heavier than those of the other servers, then the stationary distribution of the response time until the completion of service at server j ⩾ i ₀ asymptotically behaves like the stationary response time distribution in an isolated single-server queue with server i ₀. Similar asymptotics are given in the case when several service time distributions are subexponential and asymptotically tail-equivalent. This result is then extended to the asymptotics of general (max,+)-linear systems associated with i.i.d. driving matrices having one (or more) dominant diagonal entry in the subexponential class. In the irreducible case, the asymptotics are surprisingly simple, in comparison with results of the same kind in the Cramér case: the asymptotics only involve the excess distribution of the dominant diagonal entry, the mean value of this entry, the intensity of the arrival process, and the Lyapunov exponent of the sequence of driving matrices. In the reducible case, asymptotics of the same kind, though somewhat more complex, are also obtained. As a direct application, we give the asymptotics of stationary response times in a class of stochastic Petri nets called event graphs. This is based on the assumption that the firing times are independent and that the tail of the firing times of one of the transitions is subexponential and heavier than those of the others. An extension of these results to nonrenewal input processes is discussed. Asymptotics of queue size processes are also considered. This revised version was published online in June 2006 with corrections to the Cover Date.     

An appointment-based service center with guaranteed service

Customers are scheduled to arrive at periodic scheduling intervals to receive service from a single server system. A customer must start receiving service within a given departure interval; if this is not the case, the system will pay a penalty and/or transfer the customer to another facility at the system's cost. A complete transient and steady-state analysis of the system is given, the optimal scheduling and departure intervals are determined, and the virtual work in the system is analyzed.