Heavy-usage asymptotic expansions for the waiting time in closed processor-sharing systems with multiple classes

1985 ◽  
Vol 17 (1) ◽  
pp. 163-185 ◽  
Author(s):  
J. A. Morrison ◽  
D. Mitra
Keyword(s):  
Waiting Time ◽  
Asymptotic Expansions ◽  
Queueing Networks ◽  
Parabolic Cylinder ◽  
Processor Sharing ◽  
Closed Queueing Networks ◽  
Second Moments ◽  
Large Numbers ◽  
Sharing Time ◽  
Leading Term

We present new results based on novel techniques for the problem of characterizing the waiting-time distribution in a class of closed queueing networks in heavy usage, which in practical terms means that the processor is utilized more than about 80 per cent. This paper extends recent work by Mitra and Morrison [10] on the same system in normal usage. The closed system has a CPU operating under the processor-sharing (‘time-slicing’) discipline and a bank of terminals. The presence of multiple job-classes allows distinctions in the user’s behavior in the terminal and in the service requirements. This work is primarily applicable to the case of large numbers of terminals. We give an effective method for calculating, for the equilibrium waiting time, the first and second moments and the leading term in the asymptotic approximation to the distribution. Our results are in the form of asymptotic expansions in inverse powers of , where N is a large parameter. The expansion coefficients depend on the classical parabolic cylinder functions.

Heavy-usage asymptotic expansions for the waiting time in closed processor-sharing systems with multiple classes

1985 ◽  
Vol 17 (01) ◽  
pp. 163-185
Author(s):  
J. A. Morrison ◽  
D. Mitra
Keyword(s):  
Waiting Time ◽  
Asymptotic Expansions ◽  
Queueing Networks ◽  
Parabolic Cylinder ◽  
Processor Sharing ◽  
Closed Queueing Networks ◽  
Second Moments ◽  
Large Numbers ◽  
Sharing Time ◽  
Leading Term

We present new results based on novel techniques for the problem of characterizing the waiting-time distribution in a class of closed queueing networks in heavy usage, which in practical terms means that the processor is utilized more than about 80 per cent. This paper extends recent work by Mitra and Morrison [10] on the same system in normal usage. The closed system has a CPU operating under the processor-sharing (‘time-slicing’) discipline and a bank of terminals. The presence of multiple job-classes allows distinctions in the user’s behavior in the terminal and in the service requirements. This work is primarily applicable to the case of large numbers of terminals. We give an effective method for calculating, for the equilibrium waiting time, the first and second moments and the leading term in the asymptotic approximation to the distribution. Our results are in the form of asymptotic expansions in inverse powers of , where N is a large parameter. The expansion coefficients depend on the classical parabolic cylinder functions.

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.

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.

scholarly journals On the Accuracy of the Generalized Gamma Approximation to Generalized Negative Binomial Random Sums

Mathematics ◽  
2021 ◽  
Vol 9 (13) ◽  
pp. 1571
Author(s):  
Irina Shevtsova ◽  
Mikhail Tselishchev
Keyword(s):  
Negative Binomial Distribution ◽  
Binomial Distribution ◽  
Law Of Large Numbers ◽  
Negative Binomial ◽  
Infinite Divisibility ◽  
Random Sums ◽  
Generalized Gamma ◽  
Second Moments ◽  
Large Numbers ◽  
Generalized Negative Binomial Distribution

We investigate the proximity in terms of zeta-structured metrics of generalized negative binomial random sums to generalized gamma distribution with the corresponding parameters, extending thus the zeta-structured estimates of the rate of convergence in the Rényi theorem. In particular, we derive upper bounds for the Kantorovich and the Kolmogorov metrics in the law of large numbers for negative binomial random sums of i.i.d. random variables with nonzero first moments and finite second moments. Our method is based on the representation of the generalized negative binomial distribution with the shape and exponent power parameters no greater than one as a mixed geometric law and the infinite divisibility of the negative binomial distribution.

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