On the <i>GI<sup>X</sup>/G/</i>∞ system

Liu, Lei; Kashyap, B. R. K.; Templeton, J. G. C.

doi:10.2307/3214550

Cited by 80 publications

(43 citation statements)

References 7 publications

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“…Then, since the probability generating function E z N s,n is finite for any z ∈ C (see [17,Theorem 5] and [22,Theorem 1] for the detailed proof), we have…”

Section: The Blocking Probability In a System With One Resource Typementioning

confidence: 99%

Asymptotic Blocking Probabilities in Loss Networks with Subexponential Demands

Lu¹,

Radovanović²

2007

J. Appl. Probab.

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The analysis of stochastic loss networks has long been of interest in computer and communications networks and is becoming important in the areas of service and information systems. In traditional settings computing the well-known Erlang formula for blocking probabilities in these systems becomes intractable for larger resource capacities. Using compound point processes to capture stochastic variability in the request process, we generalize existing models in this framework and derive simple asymptotic expressions for the blocking probabilities. In addition, we extend our model to incorporate reserving resources in advance. Although asymptotic, our experiments show an excellent match between derived formulae and simulation results even for relatively small resource capacities and relatively large values of the blocking probabilities.

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“…Then, since the probability generating function E z N s,n is finite for any z ∈ C (see [17,Theorem 5] and [22,Theorem 1] for the detailed proof), we have…”

Section: The Blocking Probability In a System With One Resource Typementioning

confidence: 99%

Asymptotic Blocking Probabilities in Loss Networks with Subexponential Demands

Lu¹,

Radovanović²

2007

J. Appl. Probab.

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“…Now, using the result obtained in (8), one can directly derive that the second term of (7) can be asymptotically upper bounded by…”

Section: Now In Order To Show That N Can Be Replaced By N (mentioning

confidence: 99%

“…Then, since the probability generating function Ez N (<C) n is finite for any z ∈ C (see [10] and [8] for the detailed proof), we have…”

Section: Now In Order To Show That N Can Be Replaced By N (mentioning

confidence: 99%

Asymptotic performance of the non-forced idle time scheduling policies in the presence of variable demand for resources

Radovanović

Stein

2008

2008 46th Annual Allerton Conference on Communication, Control, and Computing

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A fundamental scheduling problem is to schedule a set of jobs on a set of machines so that as many jobs as possible can be scheduled while respecting the resource constraints on each machine. A further complication in many computer systems is that scheduling decisions must be made on-line, that is, as soon as job arrives it must either be scheduled or rejected. Many practical systems and algorithms schedule jobs in such a way that loads on machines tend to be balanced. However, in case when job requests are highly variable, such scheduling systems do not necessarily result in a good performance. If loads are balanced then when large jobs arrive, there may not be any "holes" large enough to hold them, thereby resulting in increased queueing delays and overall job sojourn times in the system. In this paper, we focus on addressing the job assignment problem in an on-line loss system framework, i.e., when a job arrives, it gets scheduled to a particular machine using a certain scheduling policy or, if there are not enough available resources on any of the machines, the incoming job is lost. This type of scheduling is often called Non-Forced Idle Time Scheduling and includes many common policies, including Best Fit, First Fit and Worst Fit.In this paper, we use compound point processes to capture stochastic variability in the request process, we derive simple asymptotic estimates for the job loss rate in these scheduling systems. Furthermore, we derive the asymptotic lower bound for the job loss rate by comparing it to the performance of a single large machine and derive an asympotic average case competitive ratio for the class of analyzed Non-forced Idle Time Scheduling policies. Although our proofs are asymptotic, we perform experiments which show an excellent match between simulation results and theoretical performance bounds, even for relatively small resource capacities and large values of measured loss rates.

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“…Then, since the probability generating function Ez Ns,n is finite for any z ∈ C (see Theorem 1 in [21] and Theorem 5 in [17] for the detailed proof), we have…”

Section: Theoremmentioning

confidence: 99%

Asymptotic Blocking Probabilities in Loss Networks with Subexponential Demands

Lu¹,

Radovanović²

2007

Journal of Applied Probability

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The analysis of stochastic loss networks has long been of interest in computer and communications networks and is becoming important in the areas of service and information systems. In traditional settings, computing the well known Erlang formula for blocking probability in these systems becomes intractable for larger resource capacities. Using compound point processes to capture stochastic variability in the request process, we generalize existing models in this framework and derive simple asymptotic expressions for blocking probabilities. In addition, we extend our model to incorporate reserving resources in advance. Although asymptotic, our experiments show an excellent match between derived formulas and simulation results even for relatively small resource capacities and relatively large values of blocking probabilities.

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On the GI^X/G/∞ system

Abstract: By using a shot noise process, general results on system size in continuous time are given both in transient state and in steady state with discussion on some interesting results concerning special cases. System size before arrivals is also discussed.

Cited by 80 publications

References 7 publications

Asymptotic Blocking Probabilities in Loss Networks with Subexponential Demands

Asymptotic Blocking Probabilities in Loss Networks with Subexponential Demands

Asymptotic performance of the non-forced idle time scheduling policies in the presence of variable demand for resources

Asymptotic Blocking Probabilities in Loss Networks with Subexponential Demands

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On the GIX/G/∞ system

Abstract: By using a shot noise process, general results on system size in continuous time are given both in transient state and in steady state with discussion on some interesting results concerning special cases. System size before arrivals is also discussed.

Cited by 80 publications

References 7 publications

Asymptotic Blocking Probabilities in Loss Networks with Subexponential Demands

Asymptotic Blocking Probabilities in Loss Networks with Subexponential Demands

Asymptotic performance of the non-forced idle time scheduling policies in the presence of variable demand for resources

Asymptotic Blocking Probabilities in Loss Networks with Subexponential Demands

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Product

Resources

About

On the GI^X/G/∞ system