Interloss Time in Loss System

Pierpaolo Ferrante

doi:10.1155/2009/308025

Abstract

We consider the interloss times in the Erlang Loss System. Here we present the explicit form of the probability density function of the time spent between two consecutive losses in the model. This density function solves a Cauchy problem for the second-order differential equations, which was used to evaluate the corresponding laplace transform. Finally the connection between the Erlang's loss rate and the evaluated probability density function is showed.

Highlights

We treat the random variable T i representing the time spent between the ith and the i − 1 th lost unit or ith interloss time, in the M/M/1/1 loss system
The M/M/1/1 model is characterized by the Markov property of entering and exiting processes, by one service channel and by the system capacity to accommodate one customer at a time for an overview see Medhi 1, page. 77
In Emergency Medical Systems EMSs, the nearest ambulance to the accident place is called “district unit”, and it assures the best performance to the system

Summary

Introduction

We treat the random variable T i representing the time spent between the ith and the i − 1 th lost unit or ith interloss time, in the M/M/1/1 loss system. Let Pmk t be the conditional probability to lose m clients in 0, t with k customers in the system at time t 0: Pmk t P {L t m | k}, k 0, 1, 1.2 the main results found in Ferrante 4 are the explicit values of the conditional probabilities of no losses in 0, t :. 1.6 for the M/M/1/1 loss model and to find the differential equation which governs it, in order to determine the probability density functions d fT i t dt P T i < t , 1.7 with i 1, 2, . 1.15 where r is the Erlang loss rate, and λ−1 is the interarrival mean time

First Interloss Time

Interloss Mean Time