ANALYSIS OF QUEUEING MODEL WITH PRIORITY SCHEDULING BY SUPPLEMENTARY VARIABLE METHOD

In this paper we consider the MAP/G/1 queueing system with infinite capacity. In analysis, we use the supplementary variable method to derive the double transform of the queue length and the remaining service time of the customer in service (if any) in the steady state. As will be shown in this paper, our method is very simple and elegant. As a one-dimensional marginal transform of the double transform, we obtain the generating function of the queue length in the system for the MAP/G/1 queue, which is consistent with the known result.

We propose a new approach that gives a one-step computational algorithm to directly obtain the queue length distribution of an N/G/1 queueing system. The new approach is based on the supplementary variable method and the matrix–analytic method. We shall show that this approach enables us to derive the joint distribution of the queue length and the elapsed service time.

Summary For single-server, first-come-first-served queues, specified completely by the sequences of interarrival times and service times, if these sequences are stationary and if (in some sense) there is not too much dependence within or between them, it has been shown by Kingman (1962) that the waiting-time distribution is, in heavy traffic, approximately negative exponential. This result has been extended in the present paper when there are two types of customer, one type of customer having priority of service over that of the other.

In this paper we consider a multi-server queue with a near general arrival process (represented as an arbitrary state-dependent Coxian distribution), a near general state-dependent Coxian service time distribution and a possibly finite queueing room. In addition to the dependence on the current number of customers in the system, the rate of arrivals and the progress of the service may depend on each other. We propose a semi-numerical method based on the use of conditional probabilities to compute the steady-state queue length distribution in such a queueing system. Our approach is conceptually simple, easy to implement and can be applied to both infinite and finite C m /C k /c-like queues. The proposed method uses a simple fixed-point iteration. In the case of infinite queues, it avoids the need for arbitrary truncation through the use of asymptotic conditional probabilities.This preliminary study examines the computational behavior of the proposed method with a Cox-2 service distribution. Our results indicate that it is robust and performs well even when the number of servers and the coefficient of variation of the service times are relatively high. The number of iterations to attain convergence varies from low tens to several thousand. For example, we are able to solve queues with 1024 servers and the coefficients of variation of the service time and of the time between arrivals set to 4 within 1100 iterations.KeywordsMulti-server queuegeneral arrivalsgeneral service timessteady-state queue length distributionsimple efficient semi-numerical solution

In M/G/1 retrial queueing system with two types of customers and feedback, we derived the joint generating function of the number of customers in two groups by using the supplementary variable method. It is shown that our results are consistent with those already known in the literature when -k = 0(k = 1;2);‚1 = 0 or ‚2 = 0.

Analysis of a two phase queueing system with general service times

We consider a single server retrial queueing model with two types of customers, abandonment of customers and search for priority customers. Two types of customers namely type I and type II arrive according to two independent Poisson processes and upon arrival, if the server is busy, join orbits I and II respectively. Orbit I is of finite capacity while orbit II is of Infinite capacity. Retrial attempts are made by both types of customers at constant rates and each unsuccessful retrial attempt is followed by abandonment of customers from the respective orbits with some probability. Type I customer has higher priority over type II customers. Search for customers in orbit 1 is introduced to reduce their waiting time and also to minimize the loss of such customers due to abandonment. Condition for stability is established and the system state distribution is computed. Several performance measures of the system that influences the efficiency are derived and are illustrated graphically/numerically.

We analyze a discrete-time, single-server queueing system in which the length of each service period is limited. The server takes a vacation when the limit expires or the queue empties, whichever occurs first. In the former case, the preempted service is resumed after the vacation without loss or creation of any work. This system models the transmission of message frames from a station on timed-token local-area networks (for example, FDDI and IEEE 802.4 token bus). We study the process of the unfinished work and the joint process of the queue size and the remaining service time. By using the technique of discrete Fourier transforms to determine some unknown functions in the governing equations, we numerically obtain exact mean waiting times.

In the last chapter, we used Markov process models for queueing systems with Poisson arrivals and exponential service times. To model a system as a Markov process, we should be able to give complete distribution characteristics of the process beyond time t, using what we know about the process at t and changes that may occur after t, without referring back to the events before t. When arrivals are Poisson and service times are exponential, because of the memoryless property of the exponential distribution we are able to use the Markov process as a model. If the arrival rate is λ and service rate is μ, at any time point t, time to next arrival has the exponential distribution with rate λ, and if a service is in progress, the remaining service time has the exponential distribution with rate μ. If one or both of the arrival and service distributions are non-exponential, the memoryless property does not hold and a Markov model of the type discussed in the last chapter does not work. In this chapter, we discuss a method by which a Markov model can be constructed, not for all t, but for specific time points on the time axis.KeywordsMarkov ChainService TimeTransition Probability MatrixProbability Generate FunctionService Time DistributionThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Reliability analysis of M/G/1 queueing system with repairable service station of reliability series structure

We analyze a discrete-time queueing model where two types of customers, each having their own dedicated server, are accommodated in one single FCFS queue. Service times are deterministically equal to s≥1 time slots each. New customers enter the system according to a general independent arrival process, but the types of consecutive customers may be nonindependent. As a result, arriving customers may (or may not) have the tendency to cluster according to their types, which may lead to more (or less) blocking of one type by the opposite type. The paper reveals the impact of this blocking phenomenon on the achievable throughput, the (average) system content, the (average) customer delay and the (average) unfinished work. The paper extends the results of earlier work where either the service times were assumed to be constant and equal to 1 slot each, or the customers all belonged to the same class. Our results show that, in case of Poisson arrivals, for given traffic intensity, the system-content distribution is insensitive to the length (s) of the service times, but the (mean) delay and the (mean) unfinished work in the system are not. In case of bursty arrivals, we find that all the performance measures are affected by the length (s) of the service times, for given traffic intensity.

In this paper, we establish that if an interarrival time exceeds a service time with a positive probability then the queueing systemGI/G/swith a finite waiting room always has proper limiting distributions for its characteristics such as queue length, waiting time and the remaining service times of the customers being served. The result remains valid if we consider aGI/G/ssystem with bounded waiting times. A technique is also given to establish that for a system with Poisson arrivals the limiting distributions of the queueing characteristics at an epoch of arrival and at an arbitrary epoch are identical.

Input-queued scheduling algorithms are designed to improve the throughput which use the input and output terminal in the router to obtain the maximum packets. And the length of the input queue is an important target, that can decides the superiority-inferiority about the capability of the input algorithm. With the existing algorithms, the main content about the scheduling algorithm are ensuring the packets’ high priority and low level. But most of these algorithms do not consider the length of queue. In this paper, we consider the change of queue length when the packets enter the input channel at the router, and put forward a shortest queue time-slot adapting algorithm (SQTA algorithm). First of all, SQTA algorithm can base the changes of transmittion time to control the packets’ sending time interval, and this action can make SQTA algorithm match the processing power about the router, and at the same time, this algorithm can reduce the length of the packets queue. Second, SQTA algorithm can control of a phenomenon that the packets may overflow when the length of queue is full, and that also make the packets transmitted more stable in network. The experimental result shows that this scheduling algorithm can control the length of queue effectively and reduce the waiting time with packets queue, It can be shorten the remaining service time, and can improve the network operation obviously.

A single-server queueing system with a marked Markovian arrival process of heterogeneous customers is considered. Type-1 customers have limited preemptive priority over type-2 customers. There is an infinite buffer for type-2 customers and no buffer for type-1 customers. There is also a finite buffer (stock) for consumable additional items (semi-products, half-stocks, etc.) which arrive according to the Markovian arrival process. Service of a customer requires a fixed number of consumable additional items depending on the type of the customer. The service time has a phase-type distribution depending on the type of the customer. Customers in the buffer are impatient and may leave the system without service after an exponentially distributed amount of waiting time. Aiming to minimize the loss probability of type-1 customers and maximize throughput of the system, a threshold strategy of admission to service of type-2 customers is offered. Service of type-2 customer can start only if the server is idle and the number of consumable additional items in the stock exceeds the fixed threshold. Stationary distributions of the system states and the waiting time are computed. In the numerical example, we show some interesting effects and illustrate a possibility of application of the presented results for solution of optimization problems.

Matched queues with matching batch pair [formula omitted]