Stability of Single Class Queueing Networks

2002 ◽  
pp. 13-34
Author(s):  
Harold J. Kushner
Keyword(s):  
Queueing Networks ◽  
Single Class

Some distributional approximations in Markovian queueing networks

1982 ◽  
Vol 14 (03) ◽  
pp. 654-671 ◽  
Author(s):  
T. C. Brown ◽  
P. K. Pollett
Keyword(s):  
Queueing Networks ◽  
Poisson Processes ◽  
Single Class ◽  
Open Network ◽  
State Dependent ◽  
Service Rates ◽  
Distributional Approximations ◽  
Closed Network ◽  
Poisson Approximations ◽  
Markovian Queueing Networks

We consider single-class Markovian queueing networks with state-dependent service rates (the immigration processes of Whittle (1968)). The distance of customer flows from Poisson processes is estimated in both the open and closed cases. The bounds on distances lead to simple criteria for good Poisson approximations. Using the bounds, we give an asymptotic, closed network version of the ‘loop criterion' of Melamed (1979) for an open network. Approximation of two or more flows by independent Poisson processes is also studied.

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.

Some distributional approximations in Markovian queueing networks

1982 ◽  
Vol 14 (3) ◽  
pp. 654-671 ◽  
Author(s):  
T. C. Brown ◽  
P. K. Pollett
Keyword(s):  
Queueing Networks ◽  
Poisson Processes ◽  
Single Class ◽  
Open Network ◽  
State Dependent ◽  
Service Rates ◽  
Distributional Approximations ◽  
Closed Network ◽  
Poisson Approximations ◽  
Markovian Queueing Networks

We consider single-class Markovian queueing networks with state-dependent service rates (the immigration processes of Whittle (1968)). The distance of customer flows from Poisson processes is estimated in both the open and closed cases. The bounds on distances lead to simple criteria for good Poisson approximations. Using the bounds, we give an asymptotic, closed network version of the ‘loop criterion' of Melamed (1979) for an open network. Approximation of two or more flows by independent Poisson processes is also studied.

Asymptotic expansions of moments of the waiting time in closed and open processor-sharing systems with multiple job classes

1983 ◽  
Vol 15 (4) ◽  
pp. 813-839 ◽  
Author(s):  
Debasis Mitra ◽  
J. A. Morrison
Keyword(s):  
Waiting Time ◽  
Open System ◽  
Closed System ◽  
Asymptotic Expansions ◽  
Queueing Networks ◽  
Open Systems ◽  
Asymptotic Series ◽  
Processor Sharing ◽  
Single Class ◽  
Closed Systems

We present new results based on novel techniques for the problem of characterizing the waiting-time distribution in a class of queueing networks. We give effective methods for computing, for each of possibly several job-classes, the second moment of the equilibrium waiting time for closed systems as well as for open systems. Both open and closed systems have a CPU operating under the processor-sharing (‘time-slicing') discipline in which service-time requirements may depend on job-class. The closed system also includes a bank of terminals grouped according to job-classes, with the class structure allowing distinctions in the user's behavior in the terminal. In the contrasting open system, the job streams submitted to the CPU are Poisson with rate parameters dependent on job-classes.Our results are exact for the open system and, for the closed system, in the form of an asymptotic series in inverse powers of a parameter N. In fact, the result for open networks is simply the first term in the asymptotic series. For larger closed systems, the parameter N is larger and thus fewer terms of the series need be computed to achieve a desired degree of accuracy. The complexity of the calculations for the asymptotic expansions is polynomial in number of classes and, importantly, independent of the class populations. Only the results on the single-class systems, closed and open, were previously known.

On normalization constants for closed queueing networks with finite local buffers

1998 ◽  
Vol 35 (3) ◽  
pp. 600-607
Author(s):  
Ulrich A. W. Tetzlaff
Keyword(s):  
Queueing Networks ◽  
Delta Function ◽  
Partition Functions ◽  
Balance Equations ◽  
Steady State Flow ◽  
Closed Queueing Networks ◽  
Single Class ◽  
Closed Form Solutions ◽  
Flow Balance ◽  
Number Of Customers

We present new closed form solutions for partition functions used to normalize the steady-state flow balance equations of certain Markovian type queueing networks. The results focus on single class closed product form networks with state space constraints at the queueing stations. They are achieved by combining the partition function of the open network, having finite local buffers with a delta function in order to fix the number of customers in the system.

scholarly journals Ergodic Rate Control Problem for Single Class Queueing Networks

2011 ◽  
Vol 49 (4) ◽  
pp. 1570-1606 ◽  
Author(s):  
Amarjit Budhiraja ◽  
Arka P. Ghosh ◽  
Chihoon Lee
Keyword(s):  
Control Problem ◽  
Rate Control ◽  
Queueing Networks ◽  
Single Class

Asymptotic expansions of moments of the waiting time in closed and open processor-sharing systems with multiple job classes

1983 ◽  
Vol 15 (04) ◽  
pp. 813-839 ◽  
Author(s):  
Debasis Mitra ◽  
J. A. Morrison
Keyword(s):  
Waiting Time ◽  
Open System ◽  
Closed System ◽  
Asymptotic Expansions ◽  
Queueing Networks ◽  
Open Systems ◽  
Asymptotic Series ◽  
Processor Sharing ◽  
Single Class ◽  
Closed Systems

We present new results based on novel techniques for the problem of characterizing the waiting-time distribution in a class of queueing networks. We give effective methods for computing, for each of possibly several job-classes, the second moment of the equilibrium waiting time for closed systems as well as for open systems. Both open and closed systems have a CPU operating under the processor-sharing (‘time-slicing') discipline in which service-time requirements may depend on job-class. The closed system also includes a bank of terminals grouped according to job-classes, with the class structure allowing distinctions in the user's behavior in the terminal. In the contrasting open system, the job streams submitted to the CPU are Poisson with rate parameters dependent on job-classes. Our results are exact for the open system and, for the closed system, in the form of an asymptotic series in inverse powers of a parameter N. In fact, the result for open networks is simply the first term in the asymptotic series. For larger closed systems, the parameter N is larger and thus fewer terms of the series need be computed to achieve a desired degree of accuracy. The complexity of the calculations for the asymptotic expansions is polynomial in number of classes and, importantly, independent of the class populations. Only the results on the single-class systems, closed and open, were previously known.

Approximation of single-class queueing networks with downtime-induced traffic variability

2014 ◽  
Vol 53 (13) ◽  
pp. 3871-3887 ◽  
Author(s):  
Ruth Sagron ◽  
Dean Grosbard ◽  
Gad Rabinowitz ◽  
Israel Tirkel
Keyword(s):  
Queueing Networks ◽  
Single Class
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