scholarly journals Queueing systems for multiple FBM-based traffic models

2005 ◽  
Vol 46 (3) ◽  
pp. 361-377 ◽  
Author(s):  
Mihaela T. Matache ◽  
Valentin Matache
Keyword(s):  
Brownian Motion ◽  
Upper Bound ◽  
Busy Period ◽  
Queueing System ◽  
Queueing Systems ◽  
Traffic Model ◽  
Queue Size ◽  
Buffer Allocation ◽  
Traffic Models ◽  
Overflow Probability

AbstractA multiple fractional Brownian motion (FBM)-based traffic model is considered. Various lower bounds for the overflow probability of the associated queueing system are obtained. Based on a probabilistic bound for the busy period of an ATM queueing system associated with a multiple FBM-based input traffic, a minimal dynamic buffer allocation function (DBAF) is obtained and a DBAF-allocation algorithm is designed. The purpose is to create an upper bound for the queueing system associated with the traffic. This upper bound, called a DBAF, is a function of time, dynamically bouncing with the traffic. An envelope process associated with the multiple FBM-based traffic model is introduced and used to estimate the queue size of the queueing system associated with that traffic model.

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.

Queues with semi-Markovian arrivals

1967 ◽  
Vol 4 (02) ◽  
pp. 365-379 ◽  
Author(s):  
Erhan Çinlar
Keyword(s):  
Distribution Function ◽  
Markov Chain ◽  
Finite Number ◽  
Waiting Time ◽  
Busy Period ◽  
Queueing System ◽  
Interarrival Time ◽  
Single Server ◽  
Queue Size

A queueing system with a single server is considered. There are a finite number of types of customers, and the types of successive arrivals form a Markov chain. Further, the nth interarrival time has a distribution function which may depend on the types of the nth and the n–1th arrivals. The queue size, waiting time, and busy period processes are investigated. Both transient and limiting results are given.

Heavy traffic analysis of a queueing system with bounded capacity for two types of customers

1999 ◽  
Vol 36 (4) ◽  
pp. 1155-1166 ◽  
Author(s):  
David Perry ◽  
Wolfgang Stadje
Keyword(s):  
Brownian Motion ◽  
Poisson Process ◽  
Upper Bound ◽  
Queueing System ◽  
Service System ◽  
Heavy Traffic ◽  
Arrival Times ◽  
Brownian Motion With Drift ◽  
Heavy Traffic Analysis ◽  
Capacity Bound

We study a service system with a fixed upper bound for its workload and two independent inflows of customers: frequent ‘small’ ones and occasional ‘large’ ones. The workload process generated by the small customers is modelled by a Brownian motion with drift, while the arrival times of the large customers form a Poisson process and their service times are exponentially distributed. The workload process is reflected at zero and at its upper capacity bound. We derive the stationary distribution of the workload and several related quantities and compute various important characteristics of the system.

Heavy traffic analysis of a queueing system with bounded capacity for two types of customers

1999 ◽  
Vol 36 (04) ◽  
pp. 1155-1166 ◽  
Author(s):  
David Perry ◽  
Wolfgang Stadje
Keyword(s):  
Brownian Motion ◽  
Poisson Process ◽  
Upper Bound ◽  
Queueing System ◽  
Service System ◽  
Heavy Traffic ◽  
Arrival Times ◽  
Brownian Motion With Drift ◽  
Heavy Traffic Analysis ◽  
Capacity Bound

We study a service system with a fixed upper bound for its workload and two independent inflows of customers: frequent ‘small’ ones and occasional ‘large’ ones. The workload process generated by the small customers is modelled by a Brownian motion with drift, while the arrival times of the large customers form a Poisson process and their service times are exponentially distributed. The workload process is reflected at zero and at its upper capacity bound. We derive the stationary distribution of the workload and several related quantities and compute various important characteristics of the system.

scholarly journals Fractional Brownian Motion with H < 1/2 as a Limit of Scheduled Traffic

2012 ◽  
Vol 49 (03) ◽  
pp. 710-718 ◽  
Author(s):  
Victor F. Araman ◽  
Peter W. Glynn
Keyword(s):  
Brownian Motion ◽  
Limit Theorem ◽  
Fractional Brownian Motion ◽  
Heavy Traffic ◽  
Traffic Model ◽  
Traffic Models ◽  
Heavy Traffic Limit ◽  
Scheduled Traffic ◽  
Simple Class ◽  
Traffic Processes

In this paper we show that fractional Brownian motion with H &lt; ½ can arise as a limit of a simple class of traffic processes that we call ‘scheduled traffic models’. To our knowledge, this paper provides the first simple traffic model leading to fractional Brownnian motion with H &lt; ½. We also discuss some immediate implications of this result for queues fed by scheduled traffic, including a heavy-traffic limit theorem.

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.

scholarly journals Fractional Brownian Motion with H < 1/2 as a Limit of Scheduled Traffic

2012 ◽  
Vol 49 (3) ◽  
pp. 710-718 ◽  
Author(s):  
Victor F. Araman ◽  
Peter W. Glynn
Keyword(s):  
Brownian Motion ◽  
Limit Theorem ◽  
Fractional Brownian Motion ◽  
Heavy Traffic ◽  
Traffic Model ◽  
Traffic Models ◽  
Heavy Traffic Limit ◽  
Scheduled Traffic ◽  
Simple Class ◽  
Traffic Processes

In this paper we show that fractional Brownian motion with H < ½ can arise as a limit of a simple class of traffic processes that we call ‘scheduled traffic models’. To our knowledge, this paper provides the first simple traffic model leading to fractional Brownnian motion with H < ½. We also discuss some immediate implications of this result for queues fed by scheduled traffic, including a heavy-traffic limit theorem.

A queueing process with the possibility of customers becoming servers

1991 ◽  
Vol 23 (4) ◽  
pp. 957-971
Author(s):  
Wen-Jang Huang ◽  
Prem S. Puri
Keyword(s):  
Asymptotic Behavior ◽  
Joint Distribution ◽  
Queueing System ◽  
Waiting Times ◽  
Queueing Systems ◽  
Queue Size ◽  
Queueing Process

A new queueing system called G/G/{p} is introduced and studied. In this queue, unlike standard queues, the customers after being served are allowed to become servers themselves. More precisely, at the completion of his service each customer is assumed to become a server with probability p or leave the system with probability 1 – p, independent of everything else. We make some comparisons about the waiting times and queue sizes among different queueing systems. We also study the joint distribution of the queue size, the number of servers and the number of departures at time t for exact and asymptotic behavior for large t.

New results in the theory of repeated orders queueing systems

1979 ◽  
Vol 16 (03) ◽  
pp. 631-640 ◽  
Author(s):  
Qui Hoon Choo ◽  
Brian Conolly
Keyword(s):  
Busy Period ◽  
Queueing System ◽  
Queueing Systems ◽  
Random Order ◽  
Computing System ◽  
Idle Time ◽  
Random Intervals ◽  
First Arrival ◽  
Normal Sense ◽  
System Time

The repeated orders queueing system (ROO) permits no waiting or queue in the normal sense. Instead customers who find the service (or device, to use an engineering term) busy make reapplications at random intervals and in random order until their needs are met. Thus a second demand stream supplements the basic first arrival stream. Familiar examples are provided in a telephone communication setting, in particular in the context of a multiaccess computing system. Cohen [3] and Aleksandrov [1] made the first contributions to the theory of ROO. This paper complements their work with a steady-state analysis of system time (waiting time including service of a new arrival), of service idle time, and of system busy period.

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