Invariance of the mean number of customers in priority queueing systems with a finite source

Cybernetics ◽  
1988 ◽  
Vol 23 (5) ◽  
pp. 715-719
Author(s):  
B. B. Timofeev ◽  
E. A. Semeshko
Keyword(s):  
Queueing Systems ◽  
Finite Source ◽  
The Mean ◽  
Number Of Customers ◽  
Priority Queueing

scholarly journals Bounds on the rate of convergence for one class of inhomogeneous Markovian queueing models with possible batch arrivals and services

2018 ◽  
Vol 28 (1) ◽  
pp. 141-154 ◽  
Author(s):  
Alexander Zeifman ◽  
Rostislav Razumchik ◽  
Yacov Satin ◽  
Ksenia Kiseleva ◽  
Anna Korotysheva ◽  
...  
Keyword(s):  
Rate Of Convergence ◽  
Queueing Systems ◽  
Approximation Error ◽  
Linear Operators ◽  
Upper And Lower Bounds ◽  
Logarithmic Norm ◽  
Batch Arrivals ◽  
State Dependent ◽  
The Mean ◽  
Number Of Customers

AbstractIn this paper we present a method for the computation of convergence bounds for four classes of multiserver queueing systems, described by inhomogeneous Markov chains. Specifically, we consider an inhomogeneous M/M/S queueing system with possible state-dependent arrival and service intensities, and additionally possible batch arrivals and batch service. A unified approach based on a logarithmic norm of linear operators for obtaining sharp upper and lower bounds on the rate of convergence and corresponding sharp perturbation bounds is described. As a side effect, we show, by virtue of numerical examples, that the approach based on a logarithmic norm can also be used to approximate limiting characteristics (the idle probability and the mean number of customers in the system) of the systems considered with a given approximation error.

scholarly journals Algorithmic Analysis of Finite-Source Multi-Server Heterogeneous Queueing Systems

Mathematics ◽  
2021 ◽  
Vol 9 (20) ◽  
pp. 2624
Author(s):  
Dmitry Efrosinin ◽  
Natalia Stepanova ◽  
Janos Sztrik
Keyword(s):  
Optimal Allocation ◽  
Queueing Systems ◽  
Mean Value ◽  
Threshold Policy ◽  
Heterogeneous Model ◽  
Finite Source ◽  
Long Run ◽  
The Matrix ◽  
Number Of Customers ◽  
Main Model

The paper deals with a finite-source queueing system serving one class of customers and consisting of heterogeneous servers with unequal service intensities and of one common queue. The main model has a non-preemptive service when the customer can not change the server during its service time. The optimal allocation problem is formulated as a Markov-decision one. We show numerically that the optimal policy which minimizes the long-run average number of customers in the system has a threshold structure. We derive the matrix expressions for performance measures of the system and compare the main model with alternative simplified queuing systems which are analysed for the arbitrary number of servers. We observe that the preemptive heterogeneous model operating under a threshold policy is a good approximation for the main model by calculating the mean number of customers in the system. Moreover, using the preemptive and non-preemptive queueing models with the faster server first policy the lower and upper bounds are calculated for this mean value.

On a tandem queueing model with identical service times at both counters, II

1979 ◽  
Vol 11 (3) ◽  
pp. 644-659 ◽  
Author(s):  
O. J. Boxma
Keyword(s):  
Heavy Traffic ◽  
Queueing Systems ◽  
Single Server ◽  
Service Time Distribution ◽  
In Series ◽  
Mean And Variance ◽  
Queues In Series ◽  
The Mean ◽  
Service Times ◽  
Practical Implications

This paper is devoted to the practical implications of the theoretical results obtained in Part I [1] for queueing systems consisting of two single-server queues in series in which the service times of an arbitrary customer at both queues are identical. For this purpose some tables and graphs are included. A comparison is made—mainly by numerical and asymptotic techniques—between the following two phenomena: (i) the queueing behaviour at the second counter of the two-stage tandem queue and (ii) the queueing behaviour at a single-server queue with the same offered (Poisson) traffic as the first counter and the same service-time distribution as the second counter. This comparison makes it possible to assess the influence of the first counter on the queueing behaviour at the second counter. In particular we note that placing the first counter in front of the second counter in heavy traffic significantly reduces both the mean and variance of the total time spent in the second system.

The cyclic queue with one general and one exponential server

1983 ◽  
Vol 15 (04) ◽  
pp. 857-873 ◽  
Author(s):  
O. J. Boxma
Keyword(s):  
Time Distribution ◽  
Joint Distribution ◽  
Present Model ◽  
Response Times ◽  
Strong Connection ◽  
Waiting Room ◽  
Successive Response ◽  
Proportional Rate ◽  
The Mean ◽  
Number Of Customers

This paper considers the two-stage cyclic queueing model consisting of one general (G) and one exponential (M) server. The strong connection between the present model and the M/G/1 model (with finite waiting room) is exploited to yield the joint distribution of the successive response times of a customer at the G queue and the M queue. This result reveals a surprising phenomenon: in general there is a difference between the joint distribution of the two successive response times at (first) the G queue and (then) the M queue, and the joint distribution of the two successive response times at (first) the M queue and (then) the G queue. Another associated result is an expression for the cycle-time distribution. Special consideration is given to the case that the number of customers in the system tends to ∞, while the mean service times tend to 0 at an inversely proportional rate.

The distribution of the maximum number of customers present simultaneously during a busy period for the queueing systems M/G/1 and G/M/1

1967 ◽  
Vol 4 (01) ◽  
pp. 162-179 ◽  
Author(s):  
J. W. Cohen
Keyword(s):  
Busy Period ◽  
Transition Probabilities ◽  
Queueing Systems ◽  
Entrance Time ◽  
Distribution Of The Maximum ◽  
Number Of Customers

The distribution of the maximum number of customers simultaneously present during a busy period is studied for the queueing systems M/G/1 and G/M/1. These distributions are obtained by using taboo probabilities. Some new relations for transition probabilities and entrance time distributions are derived.

Realization probability in closed Jackson queueing networks and its application

1987 ◽  
Vol 19 (03) ◽  
pp. 708-738 ◽  
Author(s):  
X. R. Cao
Keyword(s):  
Steady State ◽  
Perturbation Analysis ◽  
Queueing Networks ◽  
Sample Path ◽  
Queueing Network ◽  
Jackson Network ◽  
The Mean ◽  
Service Times ◽  
A New Technique ◽  
Number Of Customers

Perturbation analysis is a new technique which yields the sensitivities of system performance measures with respect to parameters based on one sample path of a system. This paper provides some theoretical analysis for this method. A new notion, the realization probability of a perturbation in a closed queueing network, is studied. The elasticity of the expected throughput in a closed Jackson network with respect to the mean service times can be expressed in terms of the steady-state probabilities and realization probabilities in a very simple way. The elasticity of the throughput with respect to the mean service times when the service distributions are perturbed to non-exponential distributions can also be obtained using these realization probabilities. It is proved that the sample elasticity of the throughput obtained by perturbation analysis converges to the elasticity of the expected throughput in steady-state both in mean and with probability 1 as the number of customers served goes to This justifies the existing algorithms based on perturbation analysis which efficiently provide the estimates of elasticities in practice.

scholarly journals On the ergodic distribution of oscillating queueing systems

2003 ◽  
Vol 16 (4) ◽  
pp. 311-326 ◽  
Author(s):  
Mykola Bratiychuk ◽  
Andrzej Chydzinski
Keyword(s):  
Queueing Systems ◽  
Numerical Example ◽  
New Class ◽  
Ergodic Distribution ◽  
Number Of Customers ◽  
Special Case

This paper examines a new class of queueing systems and proves a theorem on the existence of the ergodic distribution of the number of customers in such a system. An ergodic distribution is computed explicitly for the special case of a G/M−M/1 system, where the interarrival distribution does not change and both service distributions are exponential. A numerical example is also given.

scholarly journals Concavity of the throughput of tandem queueing systems with finite buffer storage space

1990 ◽  
Vol 22 (03) ◽  
pp. 764-767 ◽  
Author(s):  
Ludolf E. Meester ◽  
J. George Shanthikumar
Keyword(s):  
Queueing System ◽  
Queueing Systems ◽  
Sample Path ◽  
Concave Function ◽  
Time Interval ◽  
Single Server ◽  
Finite Buffer ◽  
Buffer Storage ◽  
Unlimited Supply ◽  
Number Of Customers

We consider a tandem queueing system with m stages and finite intermediate buffer storage spaces. Each stage has a single server and the service times are independent and exponentially distributed. There is an unlimited supply of customers in front of the first stage. For this system we show that the number of customers departing from each of the m stages during the time interval [0, t] for any t ≧ 0 is strongly stochastically increasing and concave in the buffer storage capacities. Consequently the throughput of this tandem queueing system is an increasing and concave function of the buffer storage capacities. We establish this result using a sample path recursion for the departure processes from the m stages of the tandem queueing system, that may be of independent interest. The concavity of the throughput is used along with the reversibility property of tandem queues to obtain the optimal buffer space allocation that maximizes the throughput for a three-stage tandem queue.

Sign in / Sign up

Export Citation Format

Share Document