Realization probability in closed Jackson queueing networks and its application

1987 ◽  
Vol 19 (3) ◽  
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.

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.

Realization factors and sensitivity analysis of queueing networks with state-dependent service rates

1990 ◽  
Vol 22 (1) ◽  
pp. 178-210 ◽  
Author(s):  
Xi-Ren Cao
Keyword(s):  
Steady State ◽  
Perturbation Analysis ◽  
Queueing Networks ◽  
Sample Path ◽  
Queueing Network ◽  
Service Rate ◽  
Steady State Probability ◽  
State Dependent ◽  
Service Rates ◽  
Realization Factors

The paper studies the sensitivity of the throughput with respect to a mean service rate in a closed queueing network with exponentially distributed service requirements and state-dependent service rates. The study is based on perturbation analysis of queueing networks. A new concept, the realization factor of a perturbation, is introduced. The properties of realization factors are discussed, and a set of equations specifying the realization factors are derived. The elasticity of the steady state throughput with respect to a mean service rate equals the product of the steady state probability and the corresponding realization factor. This elasticity can be estimated by applying a perturbation analysis algorithm to a sample path of the system. The sample path elasticity of the throughput with respect to a mean service rate converges with probability 1 to the elasticity of the steady state throughput. The theory provides an analytical method of calculating the throughput sensitivity and justifies the application of perturbation analysis.

Realization factors and sensitivity analysis of queueing networks with state-dependent service rates

1990 ◽  
Vol 22 (01) ◽  
pp. 178-210 ◽  
Author(s):  
Xi-Ren Cao
Keyword(s):  
Steady State ◽  
Perturbation Analysis ◽  
Queueing Networks ◽  
Sample Path ◽  
Queueing Network ◽  
Service Rate ◽  
Steady State Probability ◽  
State Dependent ◽  
Service Rates ◽  
Realization Factors

The paper studies the sensitivity of the throughput with respect to a mean service rate in a closed queueing network with exponentially distributed service requirements and state-dependent service rates. The study is based on perturbation analysis of queueing networks. A new concept, the realization factor of a perturbation, is introduced. The properties of realization factors are discussed, and a set of equations specifying the realization factors are derived. The elasticity of the steady state throughput with respect to a mean service rate equals the product of the steady state probability and the corresponding realization factor. This elasticity can be estimated by applying a perturbation analysis algorithm to a sample path of the system. The sample path elasticity of the throughput with respect to a mean service rate converges with probability 1 to the elasticity of the steady state throughput. The theory provides an analytical method of calculating the throughput sensitivity and justifies the application of perturbation analysis.

Stochastic Monotonicities in Jackson Queueing Networks

1997 ◽  
Vol 11 (1) ◽  
pp. 1-9 ◽  
Author(s):  
Torgny Lindvall
Keyword(s):  
Queueing Networks ◽  
Queueing Network ◽  
Zero Point ◽  
Monotonicity Property ◽  
Stochastic Monotonicity ◽  
Property A ◽  
Jackson Network ◽  
Queueing Process ◽  
Network Process ◽  
Number Of Customers

When starting from 0, a standard M/M/k queueing process has a second-order stochastic monotonicity property of a strong kind: its increments are stochastically decreasing (the SDI property). A first attempt to generalize this to the Jackson queueing network fails. This gives us reason to reexamine the underlying theory for stochastic monotonicity of Markov processes starting from a zero-point, in order to find a condition on a function of a Jackson network process to have the SDI property. It turns out that the total number of customers at time t has the desired property, if the network is idle at time O. We use couplings in our analysis; they are also of value in the comparison of two networks with different parameters.

Bounds on the Mean Delay in Multiclass Queueing Networks under Shortfall-Based Priority Rules

1998 ◽  
Vol 12 (3) ◽  
pp. 329-350 ◽  
Author(s):  
Sridhar Seshadri ◽  
Michael Pinedo
Keyword(s):  
Queueing Networks ◽  
Broad Class ◽  
Sample Path ◽  
Service Rate ◽  
Priority Rules ◽  
Average Delay ◽  
Multiclass Queueing Networks ◽  
Multiclass Queueing ◽  
Service Times ◽  
Number Of Customers

A significant amount of recent research has been focused on the stability of multiclass open networks of queues (MONQs). It has been shown that these networks may be unstable under various queueing disciplines even when at each one of the nodes the arrival rate is less than the service rate. Clearly, in such cases the expected delay and the expected number of customers in the system are infinite. In this paper we propose a new class of scheduling rules that can be used in multiclass queueing networks. We refer to this class as the stable shortfall-based priority (SSBP) rules. This SSBP class itself belongs to a larger class of rules, which we refer to as the shortfall-based priority (SBP) rules. SBP is a generalization of the standard non-preemptive priority rule in which customers of the same priority class are served first-come, first-served (FCFS). Rules from SBP can mimic FCFS as well as the so-called strict or head-of-the-line priority disciplines. We show that the use of any rule from the SSBP class ensures stability in a broad class of MONQs found in practice. We proceed with the construction of a sample path inequality for the work done by an SSBP server and show how this inequality can be used to derive upper bounds for the delay when service times are bounded. Bounds for the expected delay of each class of customers in an MONQ are then obtained under the assumptions that the external arrival processes have i.i.d. interarrival times, the routings are deterministic and the service times at each step of the route are bounded. In order to derive these bounds for the average delay in an MONQ we make use of some of the classical ideas of Kingman.

scholarly journals A M/M/1 Queueing Network with Classical Retrial Policy

2021 ◽  
Vol 12 (1S) ◽  
pp. 329-337
Author(s):  
S. Shanmugasundaram, Et. al.
Keyword(s):  
Steady State ◽  
Queue Length ◽  
Queueing Network ◽  
State Probability ◽  
Queueing Model ◽  
Steady State Probability ◽  
Numerical Examples ◽  
System Length ◽  
Number Of Customers ◽  
Little's Formula

In this paper we study the M/M/1 queueing model with retrial on network. We derive the steady state probability of customers in the network, the average number of customers in the all the three nodes in the system, the queue length, system length using little’s formula. The particular case is derived (no retrial). The numerical examples are given to test the correctness of the model.

Software for Queueing Analysis

2018 ◽  
pp. 255-291
Keyword(s):  
Queueing Networks ◽  
Discrete Event ◽  
Queueing Network ◽  
Queueing Analysis ◽  
Discrete Events ◽  
Technical Literature ◽  
Event Simulation ◽  
Statistical Program ◽  
Service Times ◽  
Waiting Lines

Chapter 8 gives a brief discussion of computer simulation for discrete events. The chapter lists software programs in the technical literature that outline programs for the simulation of discrete events, both of commercial origin and free programs. In addition to the lists submitted, the authors present specialized packages for analysis and simulation of waiting lines in the R language. Statistical considerations are presented, which must be taken into account when obtaining data from simulations in situations of waiting lines. Chapter 8 presents three packages of the statistical program R: the “queueing” analysis package provides versatile tools for analysis of birth- and death-based Markovian queueing models and single and multiclass product-form queueing networks; “simmer” package is a process-oriented and trajectory-based discrete-event simulation (DES) package for R; and, the purpose of the “queuecomputer” package is to calculate, deterministically, the outputs of a queueing network, given the arrival and service times of all the customers. It also uses simulation for the implementation of a method for the calculation of queues with arbitrary arrival and service times. For each theme, the authors show the use of the packages in R.

Extremal properties of the FIFO discipline in queueing networks

1992 ◽  
Vol 29 (4) ◽  
pp. 967-978 ◽  
Author(s):  
Rhonda Righter ◽  
J. George Shanthikumar
Keyword(s):  
Likelihood Ratio ◽  
Service Time ◽  
Queueing Networks ◽  
Service Rate ◽  
Service Discipline ◽  
Single Server ◽  
Single Server Queues ◽  
First Results ◽  
Service Times ◽  
Number Of Customers

We show that using the FIFO service discipline at single server stations with ILR (increasing likelihood ratio) service time distributions in networks of monotone queues results in stochastically earlier departures throughout the network. The converse is true at stations with DLR (decreasing likelihood ratio) service time distributions. We use these results to establish the validity of the following comparisons:(i) The throughput of a closed network of FIFO single-server queues will be larger (smaller) when the service times are ILR (DLR) rather than exponential with the same means.(ii) The total stationary number of customers in an open network of FIFO single-server queues with Poisson external arrivals will be stochastically smaller (larger) when the service times are ILR (DLR) rather than exponential with the same means.We also give a surprising counterexample to show that although FIFO stochastically maximizes the number of departures by any time t from an isolated single-server queue with IHR (increasing hazard rate, which is weaker than ILR) service times, this is no longer true for networks of more than one queue. Thus the ILR assumption cannot be relaxed to IHR.Finally, we consider multiclass networks of exponential single-server queues, where the class of a customer at a particular station determines its service rate at that station, and show that serving the customer with the highest service rate (which is SEPT — shortest expected processing time first) results in stochastically earlier departures throughout the network, among all preemptive work-conserving policies. We also show that a cµ rule stochastically maximizes the number of non-defective service completions by any time t when there are random, agreeable, yields.

scholarly journals Cyclic queueing networks with subexponential service times

2004 ◽  
Vol 41 (03) ◽  
pp. 791-801
Author(s):  
H. Ayhan ◽  
Z. Palmowski ◽  
S. Schlegel
Keyword(s):  
Queueing Networks ◽  
Time Distribution ◽  
Waiting Times ◽  
Queueing Network ◽  
Service Time Distribution ◽  
Tail Behavior ◽  
Cycle Times ◽  
General Service Times ◽  
Service Times ◽  
General Service

For a K-stage cyclic queueing network with N customers and general service times, we provide bounds on the nth departure time from each stage. Furthermore, we analyze the asymptotic tail behavior of cycle times and waiting times given that at least one service-time distribution is subexponential.

Calculation of sensitivities of throughputs and realization probabilities in closed queueing networks with finite buffer capacities

1989 ◽  
Vol 21 (1) ◽  
pp. 181-206 ◽  
Author(s):  
Xi-Ren Cao
Keyword(s):  
Perturbation Analysis ◽  
Queueing Networks ◽  
Queueing Network ◽  
Theoretical Background ◽  
System Throughput ◽  
Single Server ◽  
Finite Buffer ◽  
Closed Queueing Networks ◽  
Steady State Probability ◽  
Single Class

Perturbation analysis is an efficient approach to estimating the sensitivities of the performance measures of a queueing network. A new notion, called the realization probability, provides an alternative way of calculating the sensitivity of the system throughput with respect to mean service times in closed Jackson networks with single class customers and single server nodes (Cao (1987a)). This paper extends the above results to systems with finite buffer sizes. It is proved that in an indecomposable network with finite buffer sizes a perturbation will, with probability 1, be realized or lost. For systems in which no server can directly block more than one server simultaneously, the elasticity of the expected throughput can be expressed in terms of the steady state probability and the realization probability in a simple manner. The elasticity of the throughput when each customer’s service time changes by the same amount can also be calculated. These results provide some theoretical background for perturbation analysis and clarify some important issues in this area.

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