On some time-non-homogeneous diffusion approximations to queueing systems

1987 ◽  
Vol 19 (4) ◽  
pp. 974-994 ◽  
Author(s):  
V. Giorno ◽  
A. G. Nobile ◽  
L. M. Ricciardi
Keyword(s):  
Diffusion Approximation ◽  
Busy Period ◽  
Queueing Systems ◽  
Time Dependent ◽  
Single Server ◽  
Time Varying ◽  
Diffusion Approximations ◽  
Linear Drift ◽  
Transition Functions ◽  
Fcfs Discipline

Time-non-homogeneous diffusion approximations to single server–single queue–FCFS discipline systems are considered. Under various assumptions on the nature of the time-dependent functions appearing in the infinitesimal moments the transient and the regime behaviour of the approximating diffusions are analysed in some detail. Special attention is then given to the study of a diffusion approximation characterized by a linear drift and by a periodically time-varying infinitesimal variance. Unlike the behaviour of transition functions and moments, the p.d.f. of the busy period is seen to be unaffected by the presence of such periodicity.

On some time-non-homogeneous diffusion approximations to queueing systems

1987 ◽  
Vol 19 (04) ◽  
pp. 974-994 ◽  
Author(s):  
V. Giorno ◽  
A. G. Nobile ◽  
L. M. Ricciardi
Keyword(s):  
Diffusion Approximation ◽  
Busy Period ◽  
Queueing Systems ◽  
Time Dependent ◽  
Single Server ◽  
Time Varying ◽  
Diffusion Approximations ◽  
Linear Drift ◽  
Transition Functions ◽  
Fcfs Discipline

Time-non-homogeneous diffusion approximations to single server–single queue–FCFS discipline systems are considered. Under various assumptions on the nature of the time-dependent functions appearing in the infinitesimal moments the transient and the regime behaviour of the approximating diffusions are analysed in some detail. Special attention is then given to the study of a diffusion approximation characterized by a linear drift and by a periodically time-varying infinitesimal variance. Unlike the behaviour of transition functions and moments, the p.d.f. of the busy period is seen to be unaffected by the presence of such periodicity.

On a property of a refusals stream

1997 ◽  
Vol 34 (03) ◽  
pp. 800-805 ◽  
Author(s):  
Vyacheslav M. Abramov
Keyword(s):  
Waiting Time ◽  
Service Time ◽  
Busy Period ◽  
Elementary Proof ◽  
Queueing System ◽  
Queueing Systems ◽  
Single Server ◽  
Wide Family

This paper consists of two parts. The first part provides a more elementary proof of the asymptotic theorem of the refusals stream for an M/GI/1/n queueing system discussed in Abramov (1991a). The central property of the refusals stream discussed in the second part of this paper is that, if the expectations of interarrival and service time of an M/GI/1/n queueing system are equal to each other, then the expectation of the number of refusals during a busy period is equal to 1. This property is extended for a wide family of single-server queueing systems with refusals including, for example, queueing systems with bounded waiting time.

On some diffusion approximations to queueing systems

1986 ◽  
Vol 18 (4) ◽  
pp. 991-1014 ◽  
Author(s):  
V. Giorno ◽  
A. G. Nobile ◽  
L. M. Ricciardi
Keyword(s):  
Closed Form ◽  
Busy Period ◽  
Queueing System ◽  
Diffusion Processes ◽  
Queueing Systems ◽  
Diffusion Approximations ◽  
Transient Behaviour ◽  
Closed Form Solutions ◽  
Continuous Models ◽  
Reflecting Boundaries

For a class of models of adaptive queueing systems an exact diffusion approximation is derived with the aim of obtaining information on the evolution of the systems. Our approximating diffusion process includes the Wiener and the Ornstein–Uhlenbeck processes with reflecting boundaries at 0. The goodness of the approximations is thoroughly discussed and the closed-form solutions obtained for the diffusion processes are compared with those holding for the queueing system in order to investigate the conditions under which reliable information can be obtained from the approximating continuous models. For the latter the transient behaviour is quantitatively analysed and the distribution of the busy period is determined in closed form.

scholarly journals Single-server queueing systems with uniformly limited queueing time

1964 ◽  
Vol 4 (4) ◽  
pp. 489-505 ◽  
Author(s):  
D. J. Daley
Keyword(s):  
Integral Equation ◽  
Waiting Time ◽  
Time Distribution ◽  
Busy Period ◽  
Queueing System ◽  
Queueing Systems ◽  
Single Server ◽  
Waiting Time Distribution ◽  
Loss Distribution ◽  
Queueing Time

SummaryThe paper considers the queueing system GI/G/1 with a type of customer impatience, namely, that the total queueing-time is uniformly limited. Using Lindiley's approach [10], an integral equation for the limiting waiting- time distribution is derived, and this is solved explicitly for M/G/1 using an expansion of the Pollaczek-Khintchine formula. It is also solved, in principle for Ej/G/l, and explicitly for Ej/Ek/l. A duality noted between GIA(x)/GB(x)/l and GIB(x)/GA(x)/l relates solutions for GI/Ek/l to Ek/G/l. Finally the equation for the busy period in GI/G/l is derived and related to the no-customer-loss distribution and dual distributions.

Transient analysis for exponential time-limited polling models under the preemptive repeat random policy

2020 ◽  
Vol 52 (1) ◽  
pp. 32-60
Author(s):  
Roland De Haan ◽  
Ahmad Al Hanbali ◽  
Richard J. Boucherie ◽  
Jan-Kees Van Ommeren
Keyword(s):  
Queue Length ◽  
Transient Analysis ◽  
Busy Period ◽  
Queueing Systems ◽  
Polling Systems ◽  
Single Server ◽  
Direct Relation ◽  
Batch Arrivals ◽  
Exponential Time ◽  
Polling Models

AbstractPolling systems are queueing systems consisting of multiple queues served by a single server. In this paper we analyze two types of preemptive time-limited polling systems, the so-called pure and exhaustive time-limited disciplines. In particular, we derive a direct relation for the evolution of the joint queue length during the course of a server visit. The analysis of the pure time-limited discipline builds on and extends several known results for the transient analysis of an M/G/1 queue. For the analysis of the exhaustive discipline we derive several new results for the transient analysis of the M/G/1 queue during a busy period. The final expressions for both types of polling systems that we obtain generalize previous results by incorporating customer routeing, generalized service times, batch arrivals, and Markovian polling of the server.

On some diffusion approximations to queueing systems

1986 ◽  
Vol 18 (04) ◽  
pp. 991-1014 ◽  
Author(s):  
V. Giorno ◽  
A. G. Nobile ◽  
L. M. Ricciardi
Keyword(s):  
Closed Form ◽  
Busy Period ◽  
Queueing System ◽  
Diffusion Processes ◽  
Queueing Systems ◽  
Diffusion Approximations ◽  
Transient Behaviour ◽  
Closed Form Solutions ◽  
Continuous Models ◽  
Reflecting Boundaries

For a class of models of adaptive queueing systems an exact diffusion approximation is derived with the aim of obtaining information on the evolution of the systems. Our approximating diffusion process includes the Wiener and the Ornstein–Uhlenbeck processes with reflecting boundaries at 0. The goodness of the approximations is thoroughly discussed and the closed-form solutions obtained for the diffusion processes are compared with those holding for the queueing system in order to investigate the conditions under which reliable information can be obtained from the approximating continuous models. For the latter the transient behaviour is quantitatively analysed and the distribution of the busy period is determined in closed form.

On a property of a refusals stream

1997 ◽  
Vol 34 (3) ◽  
pp. 800-805 ◽  
Author(s):  
Vyacheslav M. Abramov
Keyword(s):  
Waiting Time ◽  
Service Time ◽  
Busy Period ◽  
Elementary Proof ◽  
Queueing System ◽  
Queueing Systems ◽  
Single Server ◽  
Wide Family

This paper consists of two parts. The first part provides a more elementary proof of the asymptotic theorem of the refusals stream for an M/GI/1/n queueing system discussed in Abramov (1991a). The central property of the refusals stream discussed in the second part of this paper is that, if the expectations of interarrival and service time of an M/GI/1/n queueing system are equal to each other, then the expectation of the number of refusals during a busy period is equal to 1. This property is extended for a wide family of single-server queueing systems with refusals including, for example, queueing systems with bounded waiting time.

scholarly journals A Taylor Series Approach for Service-Coupled Queueing Systems with Intermediate Load

2017 ◽  
Vol 2017 ◽  
pp. 1-10 ◽  
Author(s):  
Ekaterina Evdokimova ◽  
Sabine Wittevrongel ◽  
Dieter Fiems
Keyword(s):  
Performance Measures ◽  
Queueing Systems ◽  
Poisson Processes ◽  
Series Expansions ◽  
Service Rate ◽  
Single Server ◽  
Queueing Model ◽  
State Space Explosion ◽  
Finite Queues ◽  
Service Rates

This paper investigates the performance of a queueing model with multiple finite queues and a single server. Departures from the queues are synchronised or coupled which means that a service completion leads to a departure in every queue and that service is temporarily interrupted whenever any of the queues is empty. We focus on the numerical analysis of this queueing model in a Markovian setting: the arrivals in the different queues constitute Poisson processes and the service times are exponentially distributed. Taking into account the state space explosion problem associated with multidimensional Markov processes, we calculate the terms in the series expansion in the service rate of the stationary distribution of the Markov chain as well as various performance measures when the system is (i) overloaded and (ii) under intermediate load. Our numerical results reveal that, by calculating the series expansions of performance measures around a few service rates, we get accurate estimates of various performance measures once the load is above 40% to 50%.

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