Convexity Properties of the Erlang Loss Formula

Arie Harel

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Advanced Queuing Theory Analysis Healthcare Operations and Scheduling Optimization Optimization and Search Problems

Abstract

We prove that the throughput of the M/G/x/x system is jointly concave in the arrival and service rates. We also show that the fraction of customers lost in the M/G/x/x system is convex in the arrival rate, if the traffic intensity is below some Ρ * and concave if the traffic intensity is greater than Ρ * . For 18 or less servers, Ρ * is less than one. For 19 or more servers, Ρ * is between 1 and 1.5. Also, the fraction lost is convex in the service rate, but not jointly convex in the two rates. These results are useful in the optimal design of queueing systems.

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References

1977Fundamentals of Queueing Theory.1985Fundamentals of queueing theory (2nd ed.).1977Queueing Systems, Vol. II: Computer Applications.1977A Classified Bibliography of Research on Optimal Design and Control of Queues 1986Models for Design and Control of Stochastic, Multi-Item Batch Production Systems 1981Resource Sharing for Efficiency in Traffic Systems 1985Approximating the Equilibrium Behavior of Multi-Server Loss Systems 1987Optimal Server Allocation in a System of Multi-Server Stations

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