Analysis of overtaking in M/M/c queues

Abstract

Overtaking events can occur in parallel processing systems when customers are served simultaneously by multiple servers. We say that one customer overtakes another when the customer leaves the system after having been served ahead of another customer(s) who arrived earlier. Overtaking events are an important issue in flexible assembly systems or packet switched communication networks. Although materials (or packets) can be processed by different servers simultaneously, there is a designated order for the assembly of final products (or data). If overtaking occurs, then additional time and money may be required to rearrange materials (or packets) to the original order (called resequencing). We investigate overtaking in an M/M/c queueing system. Two distributions were considered to describe the amount of overtaking: the number of customers that an arbitrary (tagged) customer overtakes and the number of customers who overtake the tagged customer. Explicit forms of these distributions are provided for some cases. Finally, we apply our results to practical issues in flexible assembly systems and telecommunication systems.