Busy periods in time-dependent M/G/1 queues

1976 ◽  
Vol 8 (1) ◽  
pp. 195-208 ◽  
Author(s):  
Stig. I. Rosenlund
Keyword(s):  
Integral Representations ◽  
Single Server ◽  
Waiting Room ◽  
Batch Arrivals ◽  
System Of Integral Equations ◽  
Long Run ◽  
Asymptotic Expressions ◽  
In Series ◽  
Cramer’S Rule ◽  
Non Homogeneous Poisson Process

A single-server queue with batch arrivals in a non-homogeneous Poisson process and with balking is studied with respect to the busy period, using supplementary variables. A system of integral equations is obtained on the base of which the transforms are expressed in series. For the homogeneous case, assuming finite waiting room, the solutions are obtained via Cramer's rule. This gives asymptotic expressions for the expectations for large arrival intensity. An efficiency measure giving the long run loss probability is given. For a special case contour integral representations are given as solutions.

Busy periods in time-dependent M/G/1 queues

1976 ◽  
Vol 8 (01) ◽  
pp. 195-208 ◽  
Author(s):  
Stig. I. Rosenlund
Keyword(s):  
Integral Representations ◽  
Single Server ◽  
Waiting Room ◽  
Batch Arrivals ◽  
System Of Integral Equations ◽  
Long Run ◽  
Asymptotic Expressions ◽  
In Series ◽  
Cramer’S Rule ◽  
Non Homogeneous Poisson Process

A single-server queue with batch arrivals in a non-homogeneous Poisson process and with balking is studied with respect to the busy period, using supplementary variables. A system of integral equations is obtained on the base of which the transforms are expressed in series. For the homogeneous case, assuming finite waiting room, the solutions are obtained via Cramer's rule. This gives asymptotic expressions for the expectations for large arrival intensity. An efficiency measure giving the long run loss probability is given. For a special case contour integral representations are given as solutions.

An M/G/1 model with finite waiting room in which a customer remains during part of service

1973 ◽  
Vol 10 (4) ◽  
pp. 778-785 ◽  
Author(s):  
Stig I. Rosenlund
Keyword(s):  
Service System ◽  
Asymptotic Expression ◽  
Length Distribution ◽  
Loss Ratio ◽  
Efficiency Measure ◽  
Waiting Room ◽  
System Of Integral Equations ◽  
Large Intensity ◽  
Long Run ◽  
Stieltjes Transforms

An M/G/1 service system with finite waiting room is studied. A customer is served by one server in phases, during some of which a place in the waiting room is occupied. The busy period length distribution is obtained from a system of integral equations leading to a linear system in Laplace-Stieltjes transforms. An asymptotic expression, for large intensity of arrival, for the expectation of this length is given. An efficiency measure giving the long run customer loss ratio is obtained. The model is shown to apply to an inventory and a container traffic problem.

An M/G/1 model with finite waiting room in which a customer remains during part of service

1973 ◽  
Vol 10 (04) ◽  
pp. 778-785 ◽  
Author(s):  
Stig I. Rosenlund
Keyword(s):  
Service System ◽  
Asymptotic Expression ◽  
Length Distribution ◽  
Loss Ratio ◽  
Efficiency Measure ◽  
Waiting Room ◽  
System Of Integral Equations ◽  
Large Intensity ◽  
Long Run ◽  
Stieltjes Transforms

An M/G/1 service system with finite waiting room is studied. A customer is served by one server in phases, during some of which a place in the waiting room is occupied. The busy period length distribution is obtained from a system of integral equations leading to a linear system in Laplace-Stieltjes transforms. An asymptotic expression, for large intensity of arrival, for the expectation of this length is given. An efficiency measure giving the long run customer loss ratio is obtained. The model is shown to apply to an inventory and a container traffic problem.

Using Distributed-Event Parallel Simulation to Study Departures from Many Queues in Series

1993 ◽  
Vol 7 (2) ◽  
pp. 159-186 ◽  
Author(s):  
Albert G. Greenberg ◽  
Otmar Schlunk ◽  
Ward Whitt
Keyword(s):  
Transient Behavior ◽  
Parallel Simulation ◽  
General Distribution ◽  
Single Server ◽  
Long Run ◽  
Connection Machine ◽  
In Series ◽  
Parallel Prefix ◽  
First In First Out

In this paper we describe an application of distributed-event parallel simulation to study the transient behavior of a large non-Markovian network of queues. In particular, we implemented the parallel-prefix-based algorithm of Greenberg, Lubachevsky, and Mitrani [13,14] on the 8,192-processor CM-2 Connection machine and the 16,384-processor MasPar computer to simulate the departure times D(k, n) of the kth customer from the nth queue in a long series of single-server queues. Each queue has unlimited waiting space and uses the first-in first-out discipline. The service times of all the customers at all the queues are i.i.d. with a general distribution, and the system starts out with k customers in the first queue and all other queues empty. Glynn and Whitt [11] established limit theorems for this model, but very little could be said about the limits themselves. The simulation results presented here describe the limits and the quality of the approximations resulting from using the limits for finite k and n. Indeed, the simulations suggest interesting conjectures. For this model speeding up a single long run is far superior to independent replications, because very long runs are required to obtain unbiased estimates of the desired quantities and the variance of the estimator at the end of the run is small. The achieved simulation rate was about 17 billion service completions per hour, which is a speedup by about a factor of 100 compared to simulation on a conventional single-processor machine. This speedup contributed greatly to performing the desired experiments.

The interchangeability of ·/M/1 queues in series

1979 ◽  
Vol 16 (3) ◽  
pp. 690-695 ◽  
Author(s):  
Richard R. Weber
Keyword(s):  
Output Process ◽  
Input Process ◽  
Waiting Room ◽  
Series Order ◽  
In Series ◽  
Queues In Series ◽  
Stochastic Input ◽  
Pass Through ◽  
Single Exponential

A series of queues consists of a number of · /M/1 queues arranged in a series order. Each queue has an infinite waiting room and a single exponential server. The rates of the servers may differ. Initially the system is empty. Customers enter the first queue according to an arbitrary stochastic input process and then pass through the queues in order: a customer leaving the first queue immediately enters the second queue, and so on. We are concerned with the stochastic output process of customer departures from the final queue. We show that the queues are interchangeable, in the sense that the output process has the same distribution for all series arrangements of the queues. The ‘output theorem' for the M/M/1 queue is a corollary of this result.

On a tandem queueing model with identical service times at both counters, I

1979 ◽  
Vol 11 (3) ◽  
pp. 616-643 ◽  
Author(s):  
O. J. Boxma
Keyword(s):  
Waiting Times ◽  
Sufficient Conditions ◽  
Complete Analysis ◽  
Single Server ◽  
Stationary Distributions ◽  
In Series ◽  
Queues In Series ◽  
Service Times ◽  
Necessary And Sufficient ◽  
Insight Into

This paper considers a queueing system consisting of two single-server queues in series, in which the service times of an arbitrary customer at both queues are identical. Customers arrive at the first queue according to a Poisson process.Of this model, which is of importance in modern network design, a rather complete analysis will be given. The results include necessary and sufficient conditions for stationarity of the tandem system, expressions for the joint stationary distributions of the actual waiting times at both queues and of the virtual waiting times at both queues, and explicit expressions (i.e., not in transform form) for the stationary distributions of the sojourn times and of the actual and virtual waiting times at the second queue.In Part II (pp. 644–659) these results will be used to obtain asymptotic and numerical results, which will provide more insight into the general phenomenon of tandem queueing with correlated service times at the consecutive queues.

On queueing systems by retrials

1983 ◽  
Vol 20 (02) ◽  
pp. 380-389 ◽  
Author(s):  
Vidyadhar G. Kulkarni
Keyword(s):  
General Result ◽  
Service Time ◽  
Queueing Systems ◽  
Single Server ◽  
Expected Number ◽  
Waiting Room ◽  
Second Moments ◽  
Different Types ◽  
General Distributions ◽  
Arbitrary Service Time

A general result for queueing systems with retrials is presented. This result relates the expected total number of retrials conducted by an arbitrary customer to the expected total number of retrials that take place during an arbitrary service time. This result is used in the analysis of a special system where two types of customer arrive in an independent Poisson fashion at a single-server service station with no waiting room. The service times of the two types of customer have independent general distributions with finite second moments. When the incoming customer finds the server busy he immediately leaves and tries his luck again after an exponential amount of time. The retrial rates are different for different types of customers. Expressions are derived for the expected number of retrial customers of each type.

scholarly journals Large Deviations for the Single-Server Queue and the Reneging Paradox

2021 ◽  
Author(s):  
Rami Atar ◽  
Amarjit Budhiraja ◽  
Paul Dupuis ◽  
Ruoyu Wu
Keyword(s):  
Large Deviations ◽  
Decay Rate ◽  
Sample Path ◽  
Arrival Rate ◽  
Rate Function ◽  
Service Rate ◽  
Single Server ◽  
Large Deviations Principle ◽  
Long Run ◽  
The Individual

For the M/M/1+M model at the law-of-large-numbers scale, the long-run reneging count per unit time does not depend on the individual (i.e., per customer) reneging rate. This paradoxical statement has a simple proof. Less obvious is a large deviations analogue of this fact, stated as follows: the decay rate of the probability that the long-run reneging count per unit time is atypically large or atypically small does not depend on the individual reneging rate. In this paper, the sample path large deviations principle for the model is proved and the rate function is computed. Next, large time asymptotics for the reneging rate are studied for the case when the arrival rate exceeds the service rate. The key ingredient is a calculus of variations analysis of the variational problem associated with atypical reneging. A characterization of the aforementioned decay rate, given explicitly in terms of the arrival and service rate parameters of the model, is provided yielding a precise mathematical description of this paradoxical behavior.

On a tandem queueing model with identical service times at both counters, II

1979 ◽  
Vol 11 (3) ◽  
pp. 644-659 ◽  
Author(s):  
O. J. Boxma
Keyword(s):  
Heavy Traffic ◽  
Queueing Systems ◽  
Single Server ◽  
Service Time Distribution ◽  
In Series ◽  
Mean And Variance ◽  
Queues In Series ◽  
The Mean ◽  
Service Times ◽  
Practical Implications

This paper is devoted to the practical implications of the theoretical results obtained in Part I [1] for queueing systems consisting of two single-server queues in series in which the service times of an arbitrary customer at both queues are identical. For this purpose some tables and graphs are included. A comparison is made—mainly by numerical and asymptotic techniques—between the following two phenomena: (i) the queueing behaviour at the second counter of the two-stage tandem queue and (ii) the queueing behaviour at a single-server queue with the same offered (Poisson) traffic as the first counter and the same service-time distribution as the second counter. This comparison makes it possible to assess the influence of the first counter on the queueing behaviour at the second counter. In particular we note that placing the first counter in front of the second counter in heavy traffic significantly reduces both the mean and variance of the total time spent in the second system.

A Single Server Retrial Queueing System with Two Types of Batch Arrivals

2016 ◽  
pp. 55-70
Author(s):  
Kalyanaraman Rathinasabapathy
Keyword(s):  
Queueing System ◽  
Numerical Study ◽  
Priority Queue ◽  
Single Server ◽  
Type I ◽  
Type Ii ◽  
Operating Characteristics ◽  
Batch Arrivals ◽  
Retrial Queueing System ◽  
Retrial Queueing

A retrial queueing system with two types of batch arrivals is considered. The arrivals are called type I and type II customers. The type I customers arrive in batches of size k with probability c_k and type II customers arrive in batches of size k with probability d_k. Service time distributions are identical independent distributions and are different for both type of customers. If the arriving customers are blocked due to server being busy, type I customers are queued in a priority queue of infinity capacity whereas type II customers entered into retrial group in order to seek service again after a random amount of time. For this model the joint distribution of the number of customers in the priority queue and in the retrial group in closed form is obtained. Some particular models and operating characteristics are obtained. A numerical study is also carried out.

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