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.
Some sharp bounds and simple approximations are obtained for the Erlang delay formula and for the Erlang loss formula. One of the results is then used to get a simple analytical solution for the Server Allocation Problem.queues, Erlang delay formula, Erlang loss formula, inequalities, approximations, server allocation problem
We prove a strong (and seemingly odd) result about the M/M/c queue: the reciprocal of the average sojourn time is a concave function of the traffic intensity. We use this result to show that the average itself is jointly convex in arrival and service rates. The standard deviation has the same properties. Also, we determine conditions under which these properties are exhibited by a standard approximation for the M/G/c queue. These results are useful in design studies for telecommunications and production systems.
We prove some simple and sharp lower and upper bounds for the Erlang delay and loss formulae and for the number of servers that invert the Erlang delay and loss formulae. We also suggest simple and sharp approximations for the number of servers that invert the Erlang delay and loss formulae. We illustrate the importance of these bounds by using them to establish convexity proofs. We show that the probability that the M/M/s queue is empty is a decreasing and convex function of the traffic intensity. We also give a very short proof to show that the Erlang delay formula is convex in the traffic intensity when the number of servers is held constant. The complete proof of this classical result has never been published. We also give a very short proof to show that the Erlang delay formula is a convex function of the (positive integer) number of servers. One of our results is then used to get a sharp bound to the Flow Assignment Problem.Keywords Queues · Multi-server queues · Erlang delay formula · Erlang loss formulae · Erlang B formula · Erlang C formula · Bounds · Convexity · Optimal design of queue · Flow assignment problem · Number of servers · Inverse of the Erlang formulae Mathematics Subject Classification (2000) 60E15 · 60K25 · 90B22 · 90B30 · 90B50
This paper proves a long-standing conjecture regarding the optimal design of the M/M/s queue. The classical Erlang delay formula is shown to be a convex function of the number of servers when the server utilization is held constant. This means that when the server utilization is held constant, the marginal decrease in the probability that all servers are busy in the M/M/s queue brought about by the addition of two extra servers is always less than twice the decrease brought about by the addition of one extra server. As a consequence, a method of marginal analysis yields the optimal number of servers that minimize the waiting and service costs when the server utilization is held constant. In addition, it is shown that the expected number of customers in the queue and in the system, as well as the expected waiting time and sojourn in the M/M/s queue, are convex in the number of servers when the server utilization is held constant. These results are useful in design studies involving capacity planning in service operations. The classical Erlang loss formula is also shown to be a convex function of the number of servers when the server utilization is held constant.
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