scholarly journals A queue with semiperiodic traffic

2005 ◽  
Vol 37 (1) ◽  
pp. 160-184 ◽  
Author(s):  
Juan Alvarez ◽  
Bruce Hajek
Keyword(s):  
Discrete Time ◽  
Length Distribution ◽  
Arrival Rate ◽  
Arrival Process ◽  
Service Rate ◽  
Diffusion Limit ◽  
Time Analysis ◽  
Queueing Process ◽  
Brownian Bridges ◽  
Overflow Probability

In this paper, we analyze the diffusion limit of a discrete-time queueing system with constant service rate and connections that randomly enter and depart from the system. Each connection generates periodic traffic while it is active, and a connection's lifetime has finite mean. This can model a time division multiple access system with constant bit-rate connections. The diffusion scaling retains semiperiodic behavior in the limit, allowing for both short-time analysis (within one frame) and long-time analysis (over multiple frames). Weak convergence of the cumulative arrival process and the stationary buffer-length distribution is proved. It is shown that the limit of the cumulative arrival process can be viewed as a discrete-time stationary-increment Gaussian process interpolated by Brownian bridges. We present bounds on the overflow probability of the limit queueing process as functions of the arrival rate and the connection lifetime distribution. Also, numerical and simulation results are presented for geometrically distributed connection lifetimes.

scholarly journals A queue with semiperiodic traffic

2005 ◽  
Vol 37 (01) ◽  
pp. 160-184
Author(s):  
Juan Alvarez ◽  
Bruce Hajek
Keyword(s):  
Discrete Time ◽  
Length Distribution ◽  
Arrival Rate ◽  
Arrival Process ◽  
Service Rate ◽  
Diffusion Limit ◽  
Time Analysis ◽  
Queueing Process ◽  
Brownian Bridges ◽  
Overflow Probability

In this paper, we analyze the diffusion limit of a discrete-time queueing system with constant service rate and connections that randomly enter and depart from the system. Each connection generates periodic traffic while it is active, and a connection's lifetime has finite mean. This can model a time division multiple access system with constant bit-rate connections. The diffusion scaling retains semiperiodic behavior in the limit, allowing for both short-time analysis (within one frame) and long-time analysis (over multiple frames). Weak convergence of the cumulative arrival process and the stationary buffer-length distribution is proved. It is shown that the limit of the cumulative arrival process can be viewed as a discrete-time stationary-increment Gaussian process interpolated by Brownian bridges. We present bounds on the overflow probability of the limit queueing process as functions of the arrival rate and the connection lifetime distribution. Also, numerical and simulation results are presented for geometrically distributed connection lifetimes.

scholarly journals Complete analysis of a discrete-time batch service queue with batch-size-dependent service time under correlated arrival process: D-$MAP/G_n^{(a,b)}/1$

2021 ◽  
Author(s):  
Umesh Chandra Gupta ◽  
Nitin Kumar ◽  
Sourav Pradhan ◽  
Farida Parvez Barbhuiya ◽  
Mohan L Chaudhry
Keyword(s):  
Generating Function ◽  
Discrete Time ◽  
Service Time ◽  
Batch Size ◽  
Arrival Process ◽  
General Distribution ◽  
Complete Analysis ◽  
Service Rate ◽  
Single Server ◽  
Digital Platforms

Discrete-time queueing models find a large number of applications as they are used in modeling queueing systems arising in digital platforms like telecommunication systems and computer networks. In this paper, we analyze an infinite-buffer queueing model with discrete Markovian arrival process. The units on arrival are served in batches by a single server according to the general bulk-service rule, and the service time follows general distribution with service rate depending on the size of the batch being served. We mathematically formulate the model using the supplementary variable technique and obtain the vector generating function at the departure epoch. The generating function is in turn used to extract the joint distribution of queue and server content in terms of the roots of the characteristic equation. Further, we develop the relationship between the distribution at the departure epoch and the distribution at arbitrary, pre-arrival and outside observer's epochs, where the first is used to obtain the latter ones. We evaluate some essential performance measures of the system and also discuss the computing process extensively which is demonstrated by some numerical examples.

Queues with path-dependent arrival processes

2021 ◽  
Vol 58 (2) ◽  
pp. 484-504
Author(s):  
Kerry Fendick ◽  
Ward Whitt
Keyword(s):  
Point Processes ◽  
Heavy Traffic ◽  
Length Distribution ◽  
Arrival Process ◽  
Diffusion Limit ◽  
Limiting Behavior ◽  
Queue Length Distribution ◽  
Long Run ◽  
Arrival Processes ◽  
Path Dependent

AbstractWe study the transient and limiting behavior of a queue with a Pólya arrival process. The Pólya process is interesting because it exhibits path-dependent behavior, e.g. it satisfies a non-ergodic law of large numbers: the average number of arrivals over time [0, t] converges almost surely to a nondegenerate limit as $t \rightarrow \infty$. We establish a heavy-traffic diffusion limit for the $\sum_{i=1}^{n} P_i/GI/1$ queue, with arrivals occurring exogenously according to the superposition of n independent and identically distributed Pólya point processes. That limit yields a tractable approximation for the transient queue-length distribution, because the limiting net input process is a Gaussian Markov process with stationary increments. We also provide insight into the long-run performance of queues with path-dependent arrival processes. We show how Little’s law can be stated in this context, and we provide conditions under which there is stability for a queue with a Pólya arrival process.

scholarly journals Optimal control for a BMAP/G/1 queue with two service modes

1999 ◽  
Vol 5 (3) ◽  
pp. 255-273 ◽  
Author(s):  
Alexander N. Dudin ◽  
Shoichi Nishimura
Keyword(s):  
Length Distribution ◽  
Arrival Process ◽  
Service Rate ◽  
Single Server ◽  
Batch Markovian Arrival Process ◽  
Queue Length Distribution ◽  
Telecommunication Systems ◽  
Stationary Queue Length ◽  
Stationary Queue Length Distribution ◽  
Service Mode

Queueing models with controllable service rate play an important role in telecommunication systems. This paper deals with a single-server model with a batch Markovian arrival process (BMAP) and two service modes, where switch-over times are involved when changing the service mode. The embedded stationary queue length distribution and the explicit dependence of operation criteria on switch-over levels and derived.

Stationary Poisson departure processes from non-stationary queues

1986 ◽  
Vol 23 (1) ◽  
pp. 256-260 ◽  
Author(s):  
Robert D. Foley
Keyword(s):  
Poisson Process ◽  
Service Time ◽  
Queueing Systems ◽  
Arrival Rate ◽  
Distribution Functions ◽  
Arrival Process ◽  
Service Time Distribution ◽  
Departure Process ◽  
Infinite Server ◽  
Departure Processes

We present some non-stationary infinite-server queueing systems with stationary Poisson departure processes. In Foley (1982), it was shown that the departure process from the Mt/Gt/∞ queue was a Poisson process, possibly non-stationary. The Mt/Gt/∞ queue is an infinite-server queue with a stationary or non-stationary Poisson arrival process and a general server in which the service time of a customer may depend upon the customer's arrival time. Mirasol (1963) pointed out that the departure process from the M/G/∞ queue is a stationary Poisson process. The question arose whether there are any other Mt/Gt/∞ queueing systems with stationary Poisson departure processes. For example, if the arrival rate is periodic, is it possible to select the service-time distribution functions to fluctuate in order to compensate for the fluctuations of the arrival rate? In this situation and in more general situations, it is possible to select the server such that the system yields a stationary Poisson departure process.

scholarly journals Performance Modelling of Health-care Service Delivery in Adekunle Ajasin University, Akungba-Akoko, Nigeria Using Queuing Theory

2020 ◽  
pp. 119-127
Author(s):  
Orimoloye Segun Michael
Keyword(s):  
Health Care ◽  
Service Delivery ◽  
Queuing Theory ◽  
Health Care Service ◽  
Health Centre ◽  
Arrival Rate ◽  
Queuing System ◽  
Service Rate ◽  
Suitable Model ◽  
The Mean

The queuing theory is the mathematical approach to the analysis of waiting lines in any setting where arrivals rate of the subject is faster than the system can handle. It is applicable to the health care setting where the systems have excess capacity to accommodate random variation. Therefore, the purpose of this study was to determine the waiting, arrival and service times of patients at AAUA Health- setting and to model a suitable queuing system by using simulation technique to validate the model. This study was conducted at AAUA Health- Centre Akungba Akoko. It employed analytical and simulation methods to develop a suitable model. The collection of waiting time for this study was based on the arrival rate and service rate of patients at the Outpatient Centre. The data was calculated and analyzed using Microsoft Excel. Based on the analyzed data, the queuing system of the patient current situation was modelled and simulated using the PYTHON software. The result obtained from the simulation model showed that the mean arrival rate of patients on Friday week1 was lesser than the mean service rate of patients (i.e. 5.33> 5.625 (λ > µ). What this means is that the waiting line would be formed which would increase indefinitely; the service facility would always be busy. The analysis of the entire system of the AAUA health centre showed that queue length increases when the system is very busy. This work therefore evaluated and predicted the system performance of AAUA Health-Centre in terms of service delivery and propose solutions on needed resources to improve the quality of service offered to the patients visiting this health centre.

scholarly journals Large Deviations for the Single-Server Queue and the Reneging Paradox

2021 ◽  
Author(s):  
Rami Atar ◽  
Amarjit Budhiraja ◽  
Paul Dupuis ◽  
Ruoyu Wu
Keyword(s):  
Large Deviations ◽  
Decay Rate ◽  
Sample Path ◽  
Arrival Rate ◽  
Rate Function ◽  
Service Rate ◽  
Single Server ◽  
Large Deviations Principle ◽  
Long Run ◽  
The Individual

For the M/M/1+M model at the law-of-large-numbers scale, the long-run reneging count per unit time does not depend on the individual (i.e., per customer) reneging rate. This paradoxical statement has a simple proof. Less obvious is a large deviations analogue of this fact, stated as follows: the decay rate of the probability that the long-run reneging count per unit time is atypically large or atypically small does not depend on the individual reneging rate. In this paper, the sample path large deviations principle for the model is proved and the rate function is computed. Next, large time asymptotics for the reneging rate are studied for the case when the arrival rate exceeds the service rate. The key ingredient is a calculus of variations analysis of the variational problem associated with atypical reneging. A characterization of the aforementioned decay rate, given explicitly in terms of the arrival and service rate parameters of the model, is provided yielding a precise mathematical description of this paradoxical behavior.

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