Convergence rate for the distributions of GI/M/1/n and M/GI/1/n as n tends to infinity

1997 ◽  
Vol 34 (4) ◽  
pp. 1049-1060 ◽  
Author(s):  
F. Simonot
Keyword(s):  
Convergence Rate ◽  
Unique Solution ◽  
Queueing Systems ◽  
Mean Value ◽  
Traffic Intensity ◽  
Stationary Distributions ◽  
Number Of Customers

In this note, we compare the arrival and time stationary distributions of the number of customers in the GI/M/1/n and GI/M/1 queueing systems. We show that, if the inter-arrival c.d.f. H is non-lattice with mean value λ –1, and if the traffic intensity ρ = λμ –1 is strictly less than one, then the convergence rate of the stationary distributions of GI/M/1/n to the corresponding stationary distributions of GI/M/1 is geometric. More-over, the convergence rate can be characterized by the number ω, the unique solution in (0, 1) of the equation . A similar result is established for the M/GI/1/n and M/GI/1 queueing systems.

Convergence rate for the distributions of GI/M/1/n and M/GI/1/n as n tends to infinity

1997 ◽  
Vol 34 (04) ◽  
pp. 1049-1060 ◽  
Author(s):  
F. Simonot
Keyword(s):  
Convergence Rate ◽  
Unique Solution ◽  
Queueing Systems ◽  
Mean Value ◽  
Traffic Intensity ◽  
Stationary Distributions ◽  
Number Of Customers

In this note, we compare the arrival and time stationary distributions of the number of customers in the GI/M/1/n and GI/M/1 queueing systems. We show that, if the inter-arrival c.d.f. H is non-lattice with mean value λ – 1 , and if the traffic intensity ρ = λμ – 1 is strictly less than one, then the convergence rate of the stationary distributions of GI/M/1/n to the corresponding stationary distributions of GI/M/1 is geometric. More-over, the convergence rate can be characterized by the number ω, the unique solution in (0, 1) of the equation . A similar result is established for the M/GI/1/n and M/GI/1 queueing systems.

A comparison of the stationary distributions of GI/M/c/n and GI/M/c

1998 ◽  
Vol 35 (02) ◽  
pp. 510-515
Author(s):  
F. Simonot
Keyword(s):  
Unique Solution ◽  
Convergence Rates ◽  
Queueing Systems ◽  
Mean Value ◽  
Interarrival Time ◽  
Traffic Intensity ◽  
Stationary Distributions ◽  
Number Of Customers

In this note, we compare the arrival and time stationary distributions of the number of customers in the GI/M/c/n and GI/M/c queueing systems as n tends to infinity. We show that earlier results established for GI/M/1/n and GI/M/1 remain true. Namely, it is proved that if the interarrival time c.d.f. H is non lattice with mean value λ−1 and if the traffic intensity is strictly less than one, then the convergence rates in l 1 norm of the arrival and time stationary distributions of GI/M/c/n to the corresponding stationary distributions of GI/M/c are geometric and are characterized by ω, the unique solution in (0,1) of the equation z = ∫∞ 0 exp{-μc(1-z)t}dH(t).

A comparison of the stationary distributions of GI/M/c/n and GI/M/c

1998 ◽  
Vol 35 (2) ◽  
pp. 510-515 ◽  
Author(s):  
F. Simonot
Keyword(s):  
Unique Solution ◽  
Convergence Rates ◽  
Queueing Systems ◽  
Mean Value ◽  
Interarrival Time ◽  
Traffic Intensity ◽  
Stationary Distributions ◽  
Number Of Customers

In this note, we compare the arrival and time stationary distributions of the number of customers in the GI/M/c/n and GI/M/c queueing systems as n tends to infinity. We show that earlier results established for GI/M/1/n and GI/M/1 remain true. Namely, it is proved that if the interarrival time c.d.f. H is non lattice with mean value λ−1 and if the traffic intensity is strictly less than one, then the convergence rates in l1norm of the arrival and time stationary distributions of GI/M/c/n to the corresponding stationary distributions of GI/M/c are geometric and are characterized by ω, the unique solution in (0,1) of the equation z = ∫∞0 exp{-μc(1-z)t}dH(t).

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.

Fluctuations of random walk in R d and storage systems

1977 ◽  
Vol 9 (03) ◽  
pp. 566-587 ◽  
Author(s):  
Priscilla Greenwood ◽  
Moshe Shaked
Keyword(s):  
Random Walk ◽  
Unique Solution ◽  
Convex Cone ◽  
Storage Systems ◽  
Queueing Systems ◽  
Convolution Equation ◽  
Stopping Times ◽  
Multivariate Distributions ◽  
Explicit Formulas ◽  
And Storage

Two Wiener-Hopf type factorization identities for multivariate distributions are introduced. Properties of associated stopping times are derived. The structure that produces one factorization also provides the unique solution of the Wiener-Hopf convolution equation on a convex cone in R d . Some applications for multivariate storage and queueing systems are indicated. For a few cases explicit formulas are obtained for the transforms of the associated stopping times. A result of Kemperman is extended.

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.

scholarly journals On Truncation of the Matrix-Geometric Stationary Distributions

Mathematics ◽  
2019 ◽  
Vol 7 (9) ◽  
pp. 798 ◽  
Author(s):  
Naumov ◽  
Gaidamaka ◽  
Samouylov
Keyword(s):  
Finite Number ◽  
Stationary Distribution ◽  
Queueing Systems ◽  
Death Process ◽  
Birth And Death Process ◽  
Loss System ◽  
Stationary Distributions ◽  
Open Network ◽  
The Matrix

In this paper, we study queueing systems with an infinite and finite number of waiting places that can be modeled by a Quasi-Birth-and-Death process. We derive the conditions under which the stationary distribution for a loss system is a truncation of the stationary distribution of the Quasi-Birth-and-Death process and obtain the stationary distributions of both processes. We apply the obtained results to the analysis of a semi-open network in which a customer from an external queue replaces a customer leaving the system at the node from which the latter departed.

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.

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