The Economics of Segregated and Integrated Systems in Data Communication with Geometrically Distributed Message Lengths

Integrating two different message switching systems into one with the resultant integrated system having the combined channel capacity, need not necessarily result in improved performance. This segregated versus integrated system question has been adequately answered in a recent communication for the specific case when the message arrivals are Poisson processes and the message lengths for the two systems are (separately) exponentially distributed. While the Poisson assumption is probably adequate for message arrivals, recent investigations indicate that in computer communications traffic, a better approximation of the message lengths is provided by the geometric distribution. This paper presents a new algorithm which establishes the relative superiority of the segregated or the integrated system for the geometrically distributed message lengths. In order to make an equitable comparison, the average delay per message for the segregated systems is computed after making an optimum reallocation of the total available channel capacity. This is done by numerically solving a sixth-order equation. The first and second moments of the combined traffic, which need not necessarily have a geometrical distribution, are then computed. Using these statistics and the combined channel capacity, the average delay per message for the integrated system is obtained. A comparison between the performances of the segregated and the integrated systems is then effected. For situations where the segregated system might be worse than the integrated system, a finite upper bound on the ratio (average delay per message for the segregated system/average delay per message for the integrated system) exists and has been obtained. This limit is independent of the exponential or the geometric nature of the message lengths and has been established for an arbitrary number of message classes. It is also shown that for the reverse situation where the integrated system is inferior, the ratio (average delay per message for the integrated system/average delay per message for the segregated system) could be arbitrarily large. It is claimed that this knowledge imparts a useful input to the computer communication system designer in favor of the segregated system where there is relatively low confidence - in the available traffic characteristics.