Overflow Traffic from a Trunk Group with Balking

i. i Balking and Overflow Traffic A telephone call is submitted to a group of m trunks. This call may fail to occupy a trunk, even though not all m trunks are busy. There may be a number of reasons for such a failure, e.g.: the calling line may not have access to any idle trunks, some equipment other than the * This paper represents part of a doctoral dissertation submitted to the Subcommittee on Applied Mathematics, Columbia University. 45 40 T H E B E L L .SYSTEM T E C H N I C A L J O U R N A L , J A N U A R Y 19G4 trunk itself may be required to complete a connection and this equipment may be busy, or the m trunks may be merely first-stage links in a connecting network and there may be no free path through this network. Whatever the cause of the failure, we shall say that the submitted call balks (although the word is perhaps more appropriate in queueing theory applications). In this paper we shall restrict ourselves to the case in which the probability of balking depends only on the number of busy trunks: if an arriving call finds k trunks busy, it is served, with probability pk , or balks with probability qk (pk + qk = 1). If all trunks are busy, an arriving call cannot be served, and therefore q m = 1. Thus we subsume blocking under the term balking. The traffic which overflows from a trunk group with balking has different characteristics from that which overflows from a full-access group. [By a full-access trunk group we mean one for which qk = 0 (k