Steady state solution of a single-server queue with linear repeated requests

1997 ◽  
Vol 34 (1) ◽  
pp. 223-233 ◽  
Author(s):  
J. R. Artalejo ◽  
A. Gomez-Corral
Keyword(s):  
Steady State ◽  
Hypergeometric Functions ◽  
Queueing Systems ◽  
General Setting ◽  
Steady State Solution ◽  
Single Server ◽  
Service Time Distribution ◽  
Steady State Distribution ◽  
Server Queue ◽  
Number Of Customers

Queueing systems with repeated requests have many useful applications in communications and computer systems modeling. In the majority of previous work the repeat requests are made individually by each unsatisfied customer. However, there is in the literature another type of queueing situation, in which the time between two successive repeated attempts is independent of the number of customers applying for service. This paper deals with the M/G/1 queue with repeated orders in its most general setting, allowing the simultaneous presence of both types of repeat requests. We first study the steady state distribution and the partial generating functions. When the service time distribution is exponential we show that the performance characteristics can be expressed in terms of hypergeometric functions.

Steady state solution of a single-server queue with linear repeated requests

1997 ◽  
Vol 34 (01) ◽  
pp. 223-233 ◽  
Author(s):  
J. R. Artalejo ◽  
A. Gomez-Corral
Keyword(s):  
Steady State ◽  
Hypergeometric Functions ◽  
Queueing Systems ◽  
General Setting ◽  
Steady State Solution ◽  
Single Server ◽  
Service Time Distribution ◽  
Steady State Distribution ◽  
Server Queue ◽  
Number Of Customers

Queueing systems with repeated requests have many useful applications in communications and computer systems modeling. In the majority of previous work the repeat requests are made individually by each unsatisfied customer. However, there is in the literature another type of queueing situation, in which the time between two successive repeated attempts is independent of the number of customers applying for service. This paper deals with the M/G/1 queue with repeated orders in its most general setting, allowing the simultaneous presence of both types of repeat requests. We first study the steady state distribution and the partial generating functions. When the service time distribution is exponential we show that the performance characteristics can be expressed in terms of hypergeometric functions.

On a decomposition result for a class of vacation queueing systems

1990 ◽  
Vol 27 (1) ◽  
pp. 227-231 ◽  
Author(s):  
Jacqueline Loris-Teghem
Keyword(s):  
Steady State ◽  
Time Distribution ◽  
Queueing Systems ◽  
Single Server ◽  
Service Time Distribution ◽  
Steady State Distribution ◽  
State Distribution ◽  
Random Size ◽  
Poisson Arrivals ◽  
General Service

We consider a single-server infinite-capacity queueing sysem with Poisson arrivals of customer groups of random size and a general service time distribution, the server of which applies a general exhaustive service vacation policy. We are concerned with the steady-state distribution of the actual waiting time of a customer arriving while the server is active.

On a decomposition result for a class of vacation queueing systems

1990 ◽  
Vol 27 (01) ◽  
pp. 227-231 ◽  
Author(s):  
Jacqueline Loris-Teghem
Keyword(s):  
Steady State ◽  
Time Distribution ◽  
Queueing Systems ◽  
Single Server ◽  
Service Time Distribution ◽  
Steady State Distribution ◽  
State Distribution ◽  
Random Size ◽  
Poisson Arrivals ◽  
General Service

We consider a single-server infinite-capacity queueing sysem with Poisson arrivals of customer groups of random size and a general service time distribution, the server of which applies a general exhaustive service vacation policy. We are concerned with the steady-state distribution of the actual waiting time of a customer arriving while the server is active.

Transient and steady-state analysis of a queuing system having customers’ impatience with threshold

2019 ◽  
Vol 53 (5) ◽  
pp. 1861-1876 ◽  
Author(s):  
Sapana Sharma ◽  
Rakesh Kumar ◽  
Sherif Ibrahim Ammar
Keyword(s):  
Steady State ◽  
Probability Generating Function ◽  
Threshold Value ◽  
Steady State Solution ◽  
Function Technique ◽  
Single Server ◽  
Queuing Model ◽  
Queuing Models ◽  
Special Cases ◽  
Number Of Customers

In many practical queuing situations reneging and balking can only occur if the number of customers in the system is greater than a certain threshold value. Therefore, in this paper we study a single server Markovian queuing model having customers’ impatience (balking and reneging) with threshold, and retention of reneging customers. The transient analysis of the model is performed by using probability generating function technique. The expressions for the mean and variance of the number of customers in the system are obtained and a numerical example is also provided. Further the steady-state solution of the model is obtained. Finally, some important queuing models are derived as the special cases of this model.

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.

scholarly journals A single server queue under random vacation policy

2019 ◽  
Author(s):  
Priyanka kalita ◽  
Gautam Choudhury
Keyword(s):  
Steady State ◽  
Lower Cost ◽  
Queueing System ◽  
System Size ◽  
Mean Value ◽  
Single Server ◽  
Idle Period ◽  
System A ◽  
Server Queue ◽  
Number Of Customers

This paper deals with an M/G/1 queueing system with random vacation policy, in which the server takes the maximum number of random vacations till it finds minimum one message (customer) waiting in a queue at a vacation completion epoch. If no arrival occurs after completing maximum number of random vacations, the server stays dormant in the system and waits for the upcoming arrival. Here, we obtain steady state queue size distribution at an idle period completion epoch and service completion epoch. We also obtain the steady state system size probabilities and system state probabilities. Some significant measures such as a mean number of customers served during the busy period, Laplace-Stieltjes transform of unfinished work and its corresponding mean value and second moment have been obtained for the system. A cost optimal policy have been developed in terms of the average cost function to determine a locally optimal random vacation policy at a lower cost. Finally, we present various numerical results for the above system performance measures.

Constrained Load-Balancing Policies for Parallel Single-Server Queue Systems

2020 ◽  
Vol 66 (8) ◽  
pp. 3501-3527 ◽  
Author(s):  
Hung T. Do ◽  
Masha Shunko
Keyword(s):  
Load Balancing ◽  
Performance Improvement ◽  
Queueing Systems ◽  
Arrival Rate ◽  
Service Level ◽  
Single Server ◽  
Objective Functions ◽  
Expected Number ◽  
Server Queue ◽  
Number Of Customers

Flow-control policies that balance server loads are well known for improving performance of queueing systems with multiple nodes. However, although load balancing benefits the system overall, it may negatively impact some of the queueing nodes. For example, it may reduce throughput rates or engender unfairness with respect to some performance measures. For queueing systems with multiple single-server nodes, we propose a set of constrained load-balancing policies that ensures the expected arrival rate to each queueing node is not reduced, and we show that such policies provide multiple benefits for each queueing node: stochastically fewer customers and lower variance of the number of customers at each queueing node. These results imply performance improvement as measured by multiple general objective functions, including but not limited to the expected number of customers at a queueing node, probability of having a high number of customers, variance of the number of customers, and expected number of customers conditional on exceeding a threshold defined by a fixed service level. We demonstrate numerically that our proposed policies capture a large portion of the potential maximal improvement. This paper was accepted by Noah Gans, stochastic models and simulation.

scholarly journals The Queueing Model MAP|PH|1|N with Feedback Operating in a Markovian Random Environment

2016 ◽  
Vol 34 (2) ◽  
Author(s):  
A.N. Dudin ◽  
A.V. Kazimirsky ◽  
V.I. Klimenok ◽  
L. Breuer ◽  
U. Krieger
Keyword(s):  
Random Environment ◽  
Queueing Systems ◽  
Arrival Process ◽  
Single Server ◽  
Service Time Distribution ◽  
Finite Buffer ◽  
Phase Type ◽  
Server Queue ◽  
Finite State ◽  
Markovian Random Environment

Queueing systems with feedback are well suited for the description of message transmission and manufacturing processes where a repeated service is required. In the present paper we investigate a rather general single server queue with a Markovian Arrival Process (MAP), Phase-type (PH) service-time distribution, a finite buffer and feedback which operates in a random environment. A finite state Markovian random environment affects the parameters of the input and service processes and the feedback probability. The stationary distribution of the queue and of the sojourn times as well as the loss probability are calculated. Moreover, Little’s law is derived.

On the steady-state solution of the M/G/2 queue

1979 ◽  
Vol 11 (01) ◽  
pp. 240-255 ◽  
Author(s):  
Per Hokstad
Keyword(s):  
Steady State ◽  
General Solution ◽  
Laplace Transform ◽  
Queue Length ◽  
Joint Distribution ◽  
Length Distribution ◽  
Steady State Solution ◽  
Queue Length Distribution ◽  
The Difference ◽  
Number Of Customers

The asymptotic behaviour of the M/G/2 queue is studied. The difference-differential equations for the joint distribution of the number of customers present and of the remaining holding times for services in progress were obtained in Hokstad (1978a) (for M/G/m). In the present paper it is found that the general solution of these equations involves an arbitrary function. In order to decide which of the possible solutions is the answer to the queueing problem one has to consider the singularities of the Laplace transforms involved. When the service time has a rational Laplace transform, a method of obtaining the queue length distribution is outlined. For a couple of examples the explicit form of the generating function of the queue length is obtained.

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