Modelling Waiting Times for Non-Stationary Queueing Systems

2012 ◽  
Author(s):  
Mihir Dash ◽  
K. A. Venkatesh
Keyword(s):  
Waiting Times ◽  
Queueing Systems

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.

Closed queueing networks with blocking-a model for flexible manufacturing systems

1984 ◽  
Vol 16 (1) ◽  
pp. 9-9
Author(s):  
David D. W. Yao ◽  
J.A. Buzacott
Keyword(s):  
Flexible Manufacturing ◽  
Queueing Networks ◽  
Manufacturing Systems ◽  
Waiting Times ◽  
Queueing Systems ◽  
Priority Queue ◽  
Single Server ◽  
Closed Queueing Networks ◽  
State Behavior ◽  
Dynamic Priority

We consider a family of single-server queueing systems with two priority classes. The system operates under a dynamic priority queue discipline in which the relative priorities of customers increase with their waiting times, and which can be characterized by the urgency number. We investigate the transient as well as the steady-state behavior of the virtual waiting times of the two classes of customer as functions of the urgency number. Stochastic orderings, the joint distribution, and surprising limit results for these processes are obtained for the first time.

scholarly journals FINITE TWO LAYERED QUEUEING SYSTEMS

2016 ◽  
Vol 30 (3) ◽  
pp. 492-513 ◽  
Author(s):  
Efrat Perel ◽  
Uri Yechiali
Keyword(s):  
Waiting Times ◽  
Queueing Systems ◽  
Service Rate ◽  
Dimensional System ◽  
Single Server ◽  
Matrix Analytic Methods ◽  
Birth And Death Processes ◽  
Operating Modes ◽  
Loss Probabilities ◽  
Steady State Probabilities

We study layered queueing systems comprised two interlacing finite M/M/• type queues, where users of each layer are the servers of the other layer. Examples can be found in file sharing programs, SETI@home project, etc. Let Li denote the number of users in layer i, i=1, 2. We consider the following operating modes: (i) All users present in layer i join forces together to form a single server for the users in layer j (j≠i), with overall service rate μjLi (that changes dynamically as a function of the state of layer i). (ii) Each of the users present in layer i individually acts as a server for the users in layer j, with service rate μj.These operating modes lead to three different models which we analyze by formulating them as finite level-dependent quasi birth-and-death processes. We derive a procedure based on Matrix Analytic methods to derive the steady state probabilities of the two dimensional system state. Numerical examples, including mean queue sizes, mean waiting times, covariances, and loss probabilities, are presented. The models are compared and their differences are discussed.

Regenerative derivatives of regenerative sequences

1993 ◽  
Vol 25 (1) ◽  
pp. 116-139 ◽  
Author(s):  
Paul Glasserman
Keyword(s):  
Markov Chain ◽  
Steady State ◽  
Probability Space ◽  
Waiting Times ◽  
Queueing Systems ◽  
Parametric Family ◽  
Regenerative Processes ◽  
Original Sequence ◽  
Ergodic Markov Chain ◽  
Derivatives Of

Given a parametric family of regenerative processes on a common probability space, we investigate when the derivatives (with respect to the parameter) are regenerative. We primarily consider sequences satisfying explicit, Lipschitz recursions, such as the waiting times in many queueing systems, and show that derivatives regenerate together with the original sequence under reasonable monotonicity or continuity assumptions. The inputs to our recursions are i.i.d. or, more generally, governed by a Harris-ergodic Markov chain. For i.i.d. input we identify explicit regeneration points; otherwise, we use coupling arguments. We give conditions for the expected steady-state derivative to be the derivative of the steady-state mean of the original sequence. Under these conditions, the derivative of the steady-state mean has a cycle-formula representation.

A diffusion approximation for correlation in queues

1980 ◽  
Vol 17 (04) ◽  
pp. 1033-1047 ◽  
Author(s):  
C. M. Woodside ◽  
B. Pagurek ◽  
G. F. Newell
Keyword(s):  
Waiting Time ◽  
Probability Distributions ◽  
Heavy Traffic ◽  
Waiting Times ◽  
Queueing Systems ◽  
Spectral Density Function ◽  
Percentage Error ◽  
Similar Accuracy ◽  
System Time ◽  
Infinite Sum

A diffusion model is used to find heavy traffic approximate autocorrelation functions for several variables in queueing systems (i.e. for waiting time, system time, number in system and unfinished work). A table is given by which correlations can easily be found for each variable in anyGI/G/1 queue. Further, the infinite sum or integral of the autocorrelations is also found, and the spectral density function. The sum has applications in statistical analysis of queues.Extensive comparisons of approximate and exact correlations and their sums are reported, particularly for waiting times and system times, but also including number in system inM/M/1 queues. In general the correlations have similar accuracy to the probability distributions found by diffusion approximations. The percentage error is less for number in system than for waiting time.

A diffusion approximation for correlation in queues

1980 ◽  
Vol 17 (4) ◽  
pp. 1033-1047 ◽  
Author(s):  
C. M. Woodside ◽  
B. Pagurek ◽  
G. F. Newell
Keyword(s):  
Waiting Time ◽  
Probability Distributions ◽  
Heavy Traffic ◽  
Waiting Times ◽  
Queueing Systems ◽  
Spectral Density Function ◽  
Percentage Error ◽  
Similar Accuracy ◽  
System Time ◽  
Infinite Sum

A diffusion model is used to find heavy traffic approximate autocorrelation functions for several variables in queueing systems (i.e. for waiting time, system time, number in system and unfinished work). A table is given by which correlations can easily be found for each variable in any GI/G/1 queue. Further, the infinite sum or integral of the autocorrelations is also found, and the spectral density function. The sum has applications in statistical analysis of queues.Extensive comparisons of approximate and exact correlations and their sums are reported, particularly for waiting times and system times, but also including number in system in M/M/1 queues. In general the correlations have similar accuracy to the probability distributions found by diffusion approximations. The percentage error is less for number in system than for waiting time.

Queues with marked customers

1996 ◽  
Vol 28 (02) ◽  
pp. 567-587 ◽  
Author(s):  
Qi-Ming He
Keyword(s):  
Detailed Analysis ◽  
Waiting Times ◽  
Queueing Systems ◽  
Fundamental Period ◽  
Different Types ◽  
Arrival Processes ◽  
Queue Lengths ◽  
Points Of View ◽  
Markov Arrival

Queueing systems with distinguished arrivals are described on the basis of Markov arrival processes with marked transitions. Customers are distinguished by their types of arrival. Usually, the queues observed by customers of different types are different, especially for queueing systems with bursty arrival processes. We study queueing systems from the points of view of customers of different types. A detailed analysis of the fundamental period, queue lengths and waiting times at the epochs of arrivals is given. The results obtained are the generalizations of the results of theMAP/G/1 queue.

Regenerative derivatives of regenerative sequences

1993 ◽  
Vol 25 (01) ◽  
pp. 116-139 ◽  
Author(s):  
Paul Glasserman
Keyword(s):  
Markov Chain ◽  
Steady State ◽  
Probability Space ◽  
Waiting Times ◽  
Queueing Systems ◽  
Parametric Family ◽  
Regenerative Processes ◽  
Original Sequence ◽  
Ergodic Markov Chain ◽  
Derivatives Of

Given a parametric family of regenerative processes on a common probability space, we investigate when the derivatives (with respect to the parameter) are regenerative. We primarily consider sequences satisfying explicit, Lipschitz recursions, such as the waiting times in many queueing systems, and show that derivatives regenerate together with the original sequence under reasonable monotonicity or continuity assumptions. The inputs to our recursions are i.i.d. or, more generally, governed by a Harris-ergodic Markov chain. For i.i.d. input we identify explicit regeneration points; otherwise, we use coupling arguments. We give conditions for the expected steady-state derivative to be the derivative of the steady-state mean of the original sequence. Under these conditions, the derivative of the steady-state mean has a cycle-formula representation.

A solvable model for a finite-capacity queueing system

1985 ◽  
Vol 22 (04) ◽  
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.

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