Exponential Service Time Distribution Research Articles

Transient behavior of multi-server queues with recurrent input and exponential service times

Summary Customers arrive in a recurrent process and get served by one of the s (≧1) servers wit han exponential service time distribution. The equilibrium behavior of the queue length process has been studied by earlier authors. In this paper the transient behavior of this process is investigated.

Transient behavior of multi-server queues with recurrent input and exponential service times

SummaryCustomers arrive in a recurrent process and get served by one of the s (≧1) servers wit han exponential service time distribution. The equilibrium behavior of the queue length process has been studied by earlier authors. In this paper the transient behavior of this process is investigated.

An Additional Special Channel, Limited Space Queuing Problem with Service in Batches of Variable Size

This paper studies the transient behavior of a first-come-first-served queuing problem with service in batches of variable size, Poisson input, and exponential service time distribution. Further, when the queue length increases to N, a fixed pre-assigned positive number, a special service channel capable of serving a group of N units and possessing exponential service time distribution is made available and is canceled at the termination of service if the queue length becomes less than N. Also a limited waiting space K is assumed for the waiting line. The Laplace transforms of various probability generating functions are obtained and the corresponding results for the steady state are derived. Finally, some particular cases are discussed.

Queues with Hyper-Poisson Input and Exponential Service Time Distribution with State Dependent Arrival and Service Rates

The queuing system considered in this paper is characterized by (i) hyper-Poisson input with k branches with mean arrival rates depending upon the state of the system; (ii) first-come, first-served queue discipline; (iii) exponential service time distribution with mean service rate depending upon the state of the system; and (iv) finite waiting space. The functions defining the dependence of the mean arrival and service rates upon the state of the system are assumed to be arbitrary. Assuming steady-state conditions to obtain, a recurrence relation connecting the various probabilities introduced is found. By specifying a few particular forms of the state functions, graphs depicting two queue characteristics, viz., (i) probability of no delay, and (ii) mean number of units in the system are drawn. One of the particular cases has been interpreted as “a queuing problem with reneging” and another as “a multiple server queuing problem.”

Time-shared Systems

Time-shared computer (or processing) facilities are treated as stochastic queueing systems under priority service disciplines, and the performance measure of these systems is taken to be the average time spent in the system. Models are analyzed in which time-shared computer usage is obtained by giving each request a fixed quantum Q of time on the processor, after which the request is placed at the end of a queue of other requests; the queue of requests is constantly cycled, giving each user Q seconds on the machine per cycle. The case for which Q → 0 (a processor-shared model) is then analyzed using methods from queueing theory. A general time-shared facility is then considered in which priority groups are introduced. Specifically, the p th priority group is given g p Q seconds in the processor each time around. Letting Q → 0 gives results for the priority processor-shared system. These disciplines are compared with the first-come-first-served disciplines. The systems considered provide the two basic features desired in any time-shared system, namely, rapid service for short jobs and the virtual appearance of a (fractional capacity) processor available on a full-time basis. No charge is made for swap time, thus providing results for “ideal” systems. The results hold only for Poisson arrivals and geometric (or exponential) service time distributions.

On congestion systems with negative exponential desired service time distributions

On congestion systems with negative exponential desired service time distributions

Two-Server Bulk-Service Queuing Process

In the present paper a two-server queuing process fed by Poisson arrivals and exponential service time distributions has been considered under the bulk-service discipline. Time-dependent probabilities for the queue length have been obtained in terms of Laplace transforms, from which different measures associated with the queuing process could be determined. The mean queue-length and the distributions of the length of busy periods for (i) at least one channel is busy and (ii) both channels being busy, are obtained.

Some Queuing Problems with Balking and Reneging—II

This is the second of two papers in which balking (refusing to join the queue) and reneging (leaving the queue after joining) are considered. The new element here is that the balking behavior is drastically altered The model assumes (1) Customers arrive from a single infinite source in a Poisson stream (2) Arriving customers join the system if it is empty or balk with probability 1 − β/n, n = 1, 2, … where n is the number in system. Thus β is a measure of customer willingness to join the queue (3) Joining customers renege if service does not begin within a certain time. This time is a, random variable whose density function is negative exponential with parameter α (4) A single-service facility operates on a first-come, first-served basis with negative exponential service time distribution. For the steady state, the following are obtained the state probabilities, mean number in queue and system, the probability of R or more in system, the probabilities of balking, waiting, reneging, and acquiring service, the customer loss rate, the distribution and mean value of time in queue for customers who acquire service, and the corresponding results for those who renege. All of these results are also obtained for a pure balking system (no reneging) by setting the reneging parameter a equal to zero.

Asymptotic transient behaviour of the bulk service queue

Various authors have studied the transient behaviour of single-server queues. Notably, Takacs [13], [14] has analysed a queue with recurrent input and exponential service time distributions, Keilson and Kooharian [9], [10] and Finch [5] have considered a queue with general independent input and service times, Finch [6] has analysed a queue with non-recurrent input and Erlang service, and Jaiswal [8] has considered the bulk-service queue with Poisson input and Erlang service.

Some Queuing Problems with Balking and Reneging. I

Balking (refusing to join the queue) and reneging (leaving the queue after entering) are considered. The model assumes (1) Customers arrive from a single infinite source in a Poisson stream (2) Arriving customers balk with probability n/N where n is the number in system and N is the maximum number allowed in the system (3) Joining customers renege if service does not begin by a certain time, which is a random variable with negative exponential distribution (4) A single-service facility operates on a first-come, first-served basis with negative exponential service time distribution. For the steady state, the following are obtained the state probabilities, mean number in queue and system, the probability of R or more in system, the probabilities of balking, waiting, reneging, and acquiring service, the customer loss rate, the distribution and mean value of time in queue for customers who acquire service, and the corresponding results for those who renege. All of these results are also obtained for a pure balking system (no reneging) by setting the reneging parameter equal to zero. This case may be interpreted as the simple machine interference problem for which some of our results appear to be new.

Time-Dependent Solution of the Two-Server Queue Fed by General Arrival and Exponential Service Time Distributions

In this paper the queuing system GI/M/2 has been studied by using the “Phase” method. The Laplace transform of the time-dependent probability generating function and the steady-state solutions have been obtained. When the arrival time distribution is exponential, the results correspond to those obtained by Saaty (Saaty, T. L. 1960. Time-dependent solution of the many-server Poisson queue. Opns Res. 8 755–772.) for the queuing process M/M/2.

Time-Dependent Solution of the Head-Of-The-Line Priority Queue

SUMMARY The method of supplementary variables has been used to obtain the Laplace transform of the time-dependent probabilities in a head-of-the-line priority queue characterized by Poisson arrivals and general service-time distributions. An explicit solution has, however, been obtained under exponential service-time distributions with equal mean rates. The queue length probabilities, under steady-state conditions, have been evaluated and are found to be different from those obtained by Miller, thus showing that for complex queues, the asymptotic probability distribution obtained by considering the evolution of the queueing process in continuous time need not be the same as the one obtained by the imbedded Markov chain technique.

The queuing system GI/M/I with finite waiting space

Time dependent solution of the queuing system characterized by a general independent input, exponential service time distribution and a finite waiting space, has been first investigated by using the “phase method”. On finding the waiting room full, the customers then arriving may be turned away or the first customer may wait outside and the input process may be stopped till the customer then being served, completes its service. Steady state solutions of both these problems have been obtained and the difference in the operational behaviour of the two systems has been pointed out. For a 2-Erlang arrival distribution, the queuing parameters have been evaluated for different values ofρ r andN.

Delays for Last-Come First-Served Service and the Busy Period

For Poisson input to a single server, it is shown that the stationary delay distribution function for last-come first-served service is equal to the distribution function for the busy period (the interval of time during which the server is continuously busy) only for exponential distribution of service time. A similar argument shows that the identity persists for a group of fully accessible servers, each with exponential distribution of service times — a result which has been a curiosity for some time. Finally, the delay distribution for last-come first-served service and a single server with constant service time is derived.