Analysis of a non-preemptive priority multiserver queue

1988 ◽  
Vol 20 (4) ◽  
pp. 852-879 ◽  
Author(s):  
H. R. Gail ◽  
S. L. Hantler ◽  
B. A. Taylor
Keyword(s):  
Probability Distribution ◽  
Queueing System ◽  
Waiting Times ◽  
Complex Variables ◽  
Arrival Process ◽  
Preemptive Priority ◽  
Multiple Servers ◽  
The Mean ◽  
Equilibrium Probability ◽  
Number Of Customers

We consider a non-preemptive priority head of the line queueing system with multiple servers and two classes of customers. The arrival process for each class is Poisson, and the service times are exponentially distributed with different means. A Markovian state description consists of the number of customers of each class in service and in the queue. We solve a matrix equation to obtain the generating function of the equilibrium probability distribution by analyzing singularities of the equation coefficients, which are meromorphic matrices of two complex variables. We then obtain the mean waiting times for each class.

Analysis of a non-preemptive priority multiserver queue

1988 ◽  
Vol 20 (04) ◽  
pp. 852-879 ◽  
Author(s):  
H. R. Gail ◽  
S. L. Hantler ◽  
B. A. Taylor
Keyword(s):  
Probability Distribution ◽  
Queueing System ◽  
Waiting Times ◽  
Complex Variables ◽  
Arrival Process ◽  
Preemptive Priority ◽  
Multiple Servers ◽  
The Mean ◽  
Equilibrium Probability ◽  
Number Of Customers

We consider a non-preemptive priority head of the line queueing system with multiple servers and two classes of customers. The arrival process for each class is Poisson, and the service times are exponentially distributed with different means. A Markovian state description consists of the number of customers of each class in service and in the queue. We solve a matrix equation to obtain the generating function of the equilibrium probability distribution by analyzing singularities of the equation coefficients, which are meromorphic matrices of two complex variables. We then obtain the mean waiting times for each class.

scholarly journals Analysis of MAP/PH(1), PH(2)/2 Queue with Bernoulli Vacations

2008 ◽  
Vol 2008 ◽  
pp. 1-20 ◽  
Author(s):  
B. Krishna Kumar ◽  
R. Rukmani ◽  
V. Thangaraj
Keyword(s):  
Waiting Time ◽  
Time Distribution ◽  
Queueing System ◽  
Arrival Process ◽  
Waiting Time Distribution ◽  
Steady State Probability ◽  
Matrix Geometric Method ◽  
Mean Waiting Time ◽  
The Mean ◽  
Number Of Customers

We consider a two-heterogeneous-server queueing system with Bernoulli vacation in which customers arrive according to a Markovian arrival process (MAP). Servers returning from vacation immediately take another vacation if no customer is waiting. Using matrix-geometric method, the steady-state probability of the number of customers in the system is investigated. Some important performance measures are obtained. The waiting time distribution and the mean waiting time are also discussed. Finally, some numerical illustrations are provided.

A stationary Poisson departure process from a minimally delayed infinite server queue with non-stationary Poisson arrivals

1997 ◽  
Vol 34 (3) ◽  
pp. 767-772 ◽  
Author(s):  
John A. Barnes ◽  
Richard Meili
Keyword(s):  
Queueing System ◽  
Mean Shift ◽  
Intensity Function ◽  
Arrival Process ◽  
Service Time Distribution ◽  
Departure Process ◽  
Periodic Intensity ◽  
Server Queue ◽  
Poisson Arrivals ◽  
The Mean

The points of a non-stationary Poisson process with periodic intensity are independently shifted forward in time in such a way that the transformed process is stationary Poisson. The mean shift is shown to be minimal. The approach used is to consider an Mt/Gt/∞ queueing system where the arrival process is a non-stationary Poisson with periodic intensity function. A minimal service time distribution is constructed that yields a stationary Poisson departure process.

scholarly journals A discrete single server queue with Markovian arrivals and phase type group services

1995 ◽  
Vol 8 (2) ◽  
pp. 151-176 ◽  
Author(s):  
Attahiru Sule Alfa ◽  
K. Laurie Dolhun ◽  
S. Chakravarthy
Keyword(s):  
Steady State ◽  
Queueing System ◽  
State Probability ◽  
Arrival Process ◽  
Single Server ◽  
Probability Vector ◽  
Phase Type Distribution ◽  
Steady State Probability ◽  
Phase Type ◽  
Number Of Customers

We consider a single-server discrete queueing system in which arrivals occur according to a Markovian arrival process. Service is provided in groups of size no more than M customers. The service times are assumed to follow a discrete phase type distribution, whose representation may depend on the group size. Under a probabilistic service rule, which depends on the number of customers waiting in the queue, this system is studied as a Markov process. This type of queueing system is encountered in the operations of an automatic storage retrieval system. The steady-state probability vector is shown to be of (modified) matrix-geometric type. Efficient algorithmic procedures for the computation of the rate matrix, steady-state probability vector, and some important system performance measures are developed. The steady-state waiting time distribution is derived explicitly. Some numerical examples are presented.

A solvable model for a finite-capacity queueing system

1985 ◽  
Vol 22 (4) ◽  
pp. 903-911 ◽  
Author(s):  
V. Giorno ◽  
C. Negri ◽  
A. G. Nobile
Keyword(s):  
Queueing System ◽  
Waiting Times ◽  
Queueing Systems ◽  
Solvable Model ◽  
Probability Density Functions ◽  
Single Server ◽  
Finite Capacity ◽  
System Service ◽  
Transient Probabilities ◽  
Number Of Customers

Single–server–single-queue–FIFO-discipline queueing systems are considered in which at most a finite number of customers N can be present in the system. Service and arrival rates are taken to be dependent upon that state of the system. Interarrival intervals, service intervals, waiting times and busy periods are studied, and the results obtained are used to investigate the features of a special queueing model characterized by parameters (λ (Ν –n), μn). This model retains the qualitative features of the C-model proposed by Conolly [2] and Chan and Conolly [1]. However, quite unlike the latter, it also leads to closed-form expressions for the transient probabilities, the interarrival and service probability density functions and their moments, as well as the effective interarrival and service densities and their moments. Finally, some computational results are given to compare the model discussed in this paper with the C-model.

scholarly journals Markovian Queueing System with Discouraged Arrivals and Self-Regulatory Servers

2016 ◽  
Vol 2016 ◽  
pp. 1-11 ◽  
Author(s):  
K. V. Abdul Rasheed ◽  
M. Manoharan
Keyword(s):  
Queueing System ◽  
Queueing Systems ◽  
Single Server ◽  
Expected Number ◽  
Multiple Servers ◽  
Special Cases ◽  
Service Rates ◽  
Expected Waiting Time ◽  
Number Of Customers ◽  
Steady State Probabilities

We consider discouraged arrival of Markovian queueing systems whose service speed is regulated according to the number of customers in the system. We will reduce the congestion in two ways. First we attempt to reduce the congestion by discouraging the arrivals of customers from joining the queue. Secondly we reduce the congestion by introducing the concept of service switches. First we consider a model in which multiple servers have three service ratesμ1,μ2, andμ(μ1≤μ2<μ), say, slow, medium, and fast rates, respectively. If the number of customers in the system exceeds a particular pointK1orK2, the server switches to the medium or fast rate, respectively. For this adaptive queueing system the steady state probabilities are derived and some performance measures such as expected number in the system/queue and expected waiting time in the system/queue are obtained. Multiple server discouraged arrival model having one service switch and single server discouraged arrival model having one and two service switches are obtained as special cases. A Matlab program of the model is presented and numerical illustrations are given.

scholarly journals Analysis of Flexible Batch Service Queueing System to Constrict Waiting Time of Customers

2021 ◽  
Vol 2021 ◽  
pp. 1-16
Author(s):  
Deena Merit C. K. ◽  
Haridass M
Keyword(s):  
Arrival Time ◽  
Queueing System ◽  
Clinical Laboratory ◽  
Waiting Times ◽  
Distribution Center ◽  
Expiration Date ◽  
Suggested Model ◽  
Numerical Illustration ◽  
Bulk Service ◽  
Number Of Customers

When the required number of customers is available in the general bulk service (GBS) queueing system, the server begins service. Otherwise, the server will remain inactive until the number of consumers in the queue reaches that minimum required number. Customers that have already come must wait throughout this time, regardless of their arrival time. In some circumstances, like specimens awaiting testing in a clinical laboratory or perishable commodities awaiting delivery, it is necessary to finish services before the expiration date. It might only be achievable if consumers’ waiting times are kept under control. As a result, the flexible general bulk service (FGBS) rule is developed in this article to provide flexibility in batching. The effectiveness of FGBS implementation has been demonstrated using two examples: a clinical laboratory and a distribution center. To justify the suggested model, a simulation study and numerical illustration are provided.

scholarly journals Approximations in Performance Analysis of a Controllable Queueing System with Heterogeneous Servers

Mathematics ◽  
2020 ◽  
Vol 8 (10) ◽  
pp. 1803
Author(s):  
Dmitry Efrosinin ◽  
Natalia Stepanova ◽  
Janos Sztrik ◽  
Andreas Plank
Keyword(s):  
Optimal Policy ◽  
Optimal Allocation ◽  
Queueing System ◽  
Control Policy ◽  
Iteration Algorithm ◽  
Heuristic Solution ◽  
Allocation Policy ◽  
Markov Decision Model ◽  
The Mean ◽  
Number Of Customers

The paper studies a controllable multi-server heterogeneous queueing system where servers operate at different service rates without preemption, i.e., the service times are uninterrupted. The optimal control policy allocates the customers between the servers in such a way that the mean number of customers in the system reaches its minimal value. The Markov decision model and the policy-iteration algorithm are used to calculate the optimal allocation policy and corresponding mean performance characteristics. The optimal policy, when neglecting the weak influence of slow servers, is of threshold type defined as a sequence of threshold levels which specifies the queue lengths for the usage of any slower server. To avoid time-consuming calculations for systems with a large number of servers, we focus here on a heuristic evaluation of the optimal thresholds and compare this solution with the real values. We develop also the simple lower and upper bound methods based on approximation by an equivalent heterogeneous queueing system with a preemption to measure the mean number of customers in the system operating under the optimal policy. Finally, the simulation technique is used to provide sensitivity analysis of the heuristic solution to changes in the form of inter-arrival and service time distributions.

A new informative embedded Markov renewal process for the PH/G/1 queue

1986 ◽  
Vol 18 (02) ◽  
pp. 533-557 ◽  
Author(s):  
Marcel F. Neuts
Keyword(s):  
Markov Chain ◽  
Statistical Analysis ◽  
Queue Length ◽  
Busy Period ◽  
Waiting Times ◽  
Arrival Process ◽  
Embedded Markov Chain ◽  
Markov Renewal Process ◽  
The Mean ◽  
Busy Cycle

We consider a new embedded Markov chain for the PH/G/1 queue by recording the queue length, the phase of the arrival process and the number of services completed during the current busy period at the successive departure epochs. Algorithmically tractable matrix formulas are obtained which permit the analysis of the fluctuations of the queue length and waiting times during a typical busy cycle. These are useful in the computation of certain profile curves arising in the statistical analysis of queues. In addition, informative expressions for the mean waiting times in the stable GI/G/1 queue and a simple new algorithm to evaluate the waiting-time distributions for the stationary PH/PH/1 queue are obtained.

On a property of the variance of the waiting time of a queue

1968 ◽  
Vol 5 (3) ◽  
pp. 702-703 ◽  
Author(s):  
D. G. Tambouratzis
Keyword(s):  
Waiting Time ◽  
Queueing System ◽  
Mean Waiting Time ◽  
Queue Discipline ◽  
The Mean ◽  
Number Of Customers

In this note, we consider a queueing system under any discipline which does not affect the distribution of the number of customers in the queue at any time. We shall show that the variance of the waiting time is a maximum when the queue discipline is “last come, first served”. This result complements that of Kingman [1] who showed that, under the same assumptions, the mean waiting time is independent of the queue discipline and the variance of the waiting time is a minimum when the customers are served in the order of their arrival.

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