FCFS in dual-class discrete-time queueing systems

The problem with the FCFS server discipline in discrete-time queueing systems is that it doesn't actually say what happens if multiple customers enter the system at the same time, which in the discrete-time paradigm translates into 'during the same time-slot'. In other words, it doesn't specify in which order such customers are served? When we consider multiple types of customers, each requiring different service time distributions, the precise order of service even starts to affect things like queue content and delays of arbitrary customers, so specifying this order will be prime. In this paper we study a dual-class discrete-time queueing system with a general independent arrival process and generally distributed service times. The service discipline is FCFS and customers entering during the same time-slot are served in an arbitrary order. It will be our goal to search for queue content and delays of certain types of customers. If one thinks of the time-slot as a continuous but bounded time period, the arbitrary service order is equivalent to FCFS if different customers have different arrival epochs and if the arrival epochs are independent of class. For this reason we propose two distinct ways of analysing; one utilizing permutations, the other considering a bounded continuous time frame. The equivalent problem in continuous time has thus far only been solved for a Poissonian arrival process, and some specific cases of MAP.

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