The Covariance Function of a Simple Trunk Group, with Applications to Traffic Measurement*

Abstract

Erlang's classical model for telephone traffic in a loss system is considered: N trunks, calls arriving in a Poisson process and negative exponential holding times; calls which cannot be served at once are dismissed without retrials. Let N(t) be the number of trunks in use at t. An explicit formula for the covariance R(·) of N(·) in terms of the characteristic values of the transition matrix of the Markov process N ( .) is obtained. Also, R(·) is expressed purely in terms of constants and the "recovery"function, i.e. the transition probability PrIN(t) is accurately approximated by R(O)e'II, with rl the largest negative characteristic value, itself well appro:r:imated (underestimated) by -EIN(· )}/ R(O). Exact and approximate formulas for sampling error in traffic measurement are deduced from these results.

On the speed of convergence to stationarity of the Erlang loss system

“…Subsequently writing x = yD and Π = D 2 we obtain the following representation, which was first established by Beneš [1] by reference to a result in the setting of symmetric operators. (As indicated by Beneš, the representation is implied by an observation of Kramer's [13] in the setting of reversible Markov chains.…”

“…. , x N ) such that x min := min i {x i } > 0, the total variation distance between the distributions p (1) (t) and p (2) (t) in the Erlang loss model with N servers, and arrival rate λ(τ ) and service rate μ(τ ) per server at time τ, satisfies…”

“…Then for any two initial distributions p (1) (0) and p (2) (0), the total variation distance between the distributions p (1) (t) and p (2) Open Access This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.…”

Operations Research at Bell Laboratories Through the 1970s: Part I

“…Statistical aspects of the sampling techniques used in traffic measurement were analyzed by Wilkinson (1941), John Riordan (1951), Hayward (1952), Václav E. Beneš (1961), Alfred Descloux (1965Descloux ( , 1976, and others. Tables of traffic system behavior were published by Molina (1927), Riordan (1953), Wilkinson (1953Wilkinson ( , 1970, Paul J.…”

“…Optimal switching-network designs of several kinds were devised in Germany and Japan, and at Bell Labs by Alexander Feiner and Joseph G. Kappel (1970). Beneš (1965) proposed a general theoretical framework for such problems.…”