scholarly journals An Efficient Convolution Algorithm for the Non-Markovian Two-Node Cyclic Network

2021 ◽  
Vol 2021 ◽  
pp. 1-13
Author(s):  
Muhammad El-Taha ◽  
Thomas Michaud
Keyword(s):  
Service Time ◽  
Transition Probabilities ◽  
System Size ◽  
Single Server ◽  
Service Time Distribution ◽  
Birth Process ◽  
System Availability ◽  
Imbedded Markov Chain ◽  
Convolution Algorithm ◽  
Cyclic Network

Consider a closed cyclic queueing model that consists of two nodes and a total of M customers. Each node buffer can accommodate all M customers. Node 1 has N ≤ M servers, each having an exponential service time with rate λ . The second node consists of a single server with a general service time distribution function B . . The well-known machine repair model with spares, where a set of identical machines, N , is served by a single repair person, is a key application of this model. This model has several other applications in performance evaluation, manufacturing, computer networks, and in reliability studies as it can be easily used to compute system availability. In this article, we give an efficient algorithm to derive an exact solution for the steady state system size probabilities. Our approach is based on developing an efficient polynomial convolution method to compute the transition probabilities of a birth process over node 2 service times and solving an imbedded Markov chain at node 2 service completion epochs. This is a significant improvement over the exponential algorithm given in an earlier paper. Numerical examples are given to demonstrate the performance of our method.

The general bulk queue as a matrix factorisation problem of the Wiener-Hopf type. Part II.

1975 ◽  
Vol 7 (3) ◽  
pp. 647-655 ◽  
Author(s):  
John Dagsvik
Keyword(s):  
Waiting Time ◽  
Service Time ◽  
Time Distribution ◽  
Single Server ◽  
Service Time Distribution ◽  
Time Process ◽  
Waiting Time Process ◽  
Bulk Queue ◽  
Erlang Distributions ◽  
Matrix Factorisation

In a previous paper (Dagsvik (1975)) the waiting time process of the single server bulk queue is considered and a corresponding waiting time equation is established. In this paper the waiting time equation is solved when the inter-arrival or service time distribution is a linear combination of Erlang distributions. The analysis is essentially based on algebraic arguments.

The discrete-time single-server queue with time-inhomogeneous compound Poisson input and general service time distribution

1978 ◽  
Vol 15 (3) ◽  
pp. 590-601 ◽  
Author(s):  
Do Le Minh
Keyword(s):  
Discrete Time ◽  
Service Time ◽  
Time Distribution ◽  
Single Server ◽  
Service Time Distribution ◽  
Queueing Model ◽  
Compound Poisson ◽  
Poisson Input ◽  
Server Queue ◽  
General Service

This paper studies a discrete-time, single-server queueing model having a compound Poisson input with time-dependent parameters and a general service time distribution.All major transient characteristics of the system can be calculated very easily. For the queueing model with periodic arrival function, some explicit results are obtained.

On the integral of the workload process of the single server queue

2003 ◽  
Vol 40 (01) ◽  
pp. 200-225 ◽  
Author(s):  
A. A. Borovkov ◽  
O. J. Boxma ◽  
Z. Palmowski
Keyword(s):  
Service Time ◽  
Time Distribution ◽  
Busy Period ◽  
Single Server ◽  
Service Time Distribution ◽  
Whittaker Functions ◽  
Single Server Queue ◽  
Stieltjes Transform ◽  
Server Queue ◽  
Sequential Approximation

This paper is devoted to a study of the integral of the workload process of the single server queue, in particular during one busy period. Firstly, we find asymptotics of the area 𝒜 swept under the workload process W(t) during the busy period when the service time distribution has a regularly varying tail. We also investigate the case of a light-tailed service time distribution. Secondly, we consider the problem of obtaining an explicit expression for the distribution of 𝒜. In the general GI/G/1 case, we use a sequential approximation to find the Laplace—Stieltjes transform of 𝒜. In the M/M/1 case, this transform is obtained explicitly in terms of Whittaker functions. Thirdly, we consider moments of 𝒜 in the GI/G/1 queue. Finally, we show asymptotic normality of .

scholarly journals An M/G/1 queue with multiple types of feedback and gated vacations

1997 ◽  
Vol 34 (03) ◽  
pp. 773-784 ◽  
Author(s):  
Onno J. Boxma ◽  
Uri Yechiali
Keyword(s):  
Waiting Time ◽  
Service Time ◽  
Queue Length ◽  
Time Distribution ◽  
Single Server ◽  
Service Time Distribution ◽  
Waiting Time Distribution ◽  
Single Server Queue ◽  
Server Queue ◽  
Poisson Arrivals

This paper considers a single-server queue with Poisson arrivals and multiple customer feedbacks. If the first service attempt of a newly arriving customer is not successful, he returns to the end of the queue for another service attempt, with a different service time distribution. He keeps trying in this manner (as an ‘old' customer) until his service is successful. The server operates according to the ‘gated vacation' strategy; when it returns from a vacation to find K (new and old) customers, it renders a single service attempt to each of them and takes another vacation, etc. We study the joint queue length process of new and old customers, as well as the waiting time distribution of customers. Some extensions are also discussed.

M/Ck/1 Queue with Impatient Customers

2016 ◽  
pp. 22-37
Author(s):  
Umay Uzunoglu Kocer
Keyword(s):  
Service Time ◽  
Time Distribution ◽  
Random Variable ◽  
Death Process ◽  
Single Server ◽  
Service Time Distribution ◽  
Birth And Death Process ◽  
Impatient Customers ◽  
Constant Rate ◽  
Long Time

A single-server queuing system with impatient customers and Coxian service is examined. It is assumed that arrivals are Poisson with a constant rate. When the server is busy upon an arrival, customer joins the queue and there is an infinite capacity of the queue. Since the variance of the service time is relatively high, the service time distribution is characterized by k-phase Cox distribution. Due to the high variability of service times and since some of the services take extremely long time, customers not only in the queue, but also in the service may become impatient. Each customer, upon arrival, activates an individual timer and starts his patience time. The patience time for each customer is a random variable which has exponential distribution. If the service does not completed before the customer's time expires, the customer abandons the queue never to return. The model is expressed as birth-and-death process and the balance equations are provided.

Computational analysis of single-server bulk-service queues, M/GY/ 1

1989 ◽  
Vol 21 (1) ◽  
pp. 207-225 ◽  
Author(s):  
G. Brière ◽  
M. L. Chaudhry
Keyword(s):  
Service Time ◽  
Flexible Manufacturing ◽  
Time Distribution ◽  
Random Variable ◽  
Single Server ◽  
Service Time Distribution ◽  
Computationally Efficient ◽  
Uniform Distributions ◽  
Bulk Service ◽  
Number Of Customers

Algorithms are proposed for the numerical inversion of the analytical solutions obtained through classical transform methods. We compute steady-state probabilities and moments of the number of customers in the system (or in the queue) at three different epochs—postdeparture, random, and prearrival—for models of the type M/GY/1, where the capacity of the single server is a random variable. This implies first finding roots of the characteristic equation, which is detailed in an appendix for a general service time distribution. Numerical results, given a service time distribution, are illustrated through graphs and tables for cases covered in this study: deterministic, Erlang, hyperexponential, and uniform distributions. In all cases, the proposed method is computationally efficient and accurate, even for high values of the queueing parameters. The procedure is adaptable to other models in queueing theory (especially bulk queues), to problems in inventory control, transportation, flexible manufacturing process, etc. Exact results that can be obtained from the algorithms presented here will be found useful to test inequalities, bounds, or approximations.

scholarly journals Stationary queue length of a single-server queue with negative arrivals and nonexponential service time distributions

2018 ◽  
Vol 189 ◽  
pp. 02006 ◽  
Author(s):  
S K Koh ◽  
C H Chin ◽  
Y F Tan ◽  
L E Teoh ◽  
A H Pooi ◽  
...  
Keyword(s):  
Service Time ◽  
Queue Length ◽  
Queueing Systems ◽  
Length Distribution ◽  
Single Server ◽  
Service Time Distribution ◽  
Single Server Queue ◽  
Negative Customers ◽  
Stationary Queue Length ◽  
Server Queue

In this paper, a single-server queue with negative customers is considered. The arrival of a negative customer will remove one positive customer that is being served, if any is present. An alternative approach will be introduced to derive a set of equations which will be solved to obtain the stationary queue length distribution. We assume that the service time distribution tends to a constant asymptotic rate when time t goes to infinity. This assumption will allow for finding the stationary queue length of queueing systems with non-exponential service time distributions. Numerical examples for gamma distributed service time with fractional value of shape parameter will be presented in which the steady-state distribution of queue length with such service time distributions may not be easily computed by most of the existing analytical methods.

The discrete-time single-server queue with time-inhomogeneous compound Poisson input and general service time distribution

1978 ◽  
Vol 15 (03) ◽  
pp. 590-601 ◽  
Author(s):  
Do Le Minh
Keyword(s):  
Discrete Time ◽  
Service Time ◽  
Time Distribution ◽  
Single Server ◽  
Service Time Distribution ◽  
Queueing Model ◽  
Compound Poisson ◽  
Poisson Input ◽  
Server Queue ◽  
General Service

This paper studies a discrete-time, single-server queueing model having a compound Poisson input with time-dependent parameters and a general service time distribution. All major transient characteristics of the system can be calculated very easily. For the queueing model with periodic arrival function, some explicit results are obtained.

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