4.9.4 Preemptive Priorities

      
        Some results also exist for the case of systems with preemptive priorities, both for the preemptive-resume and the preemptive-repeat types. The former refers to the situation in which service to an ejected user, once that user regains access to a server, continues from the point where it was interrupted earlier. By contrast, in the latter type of preemptive system, all service received already is "lost" once a user is ejected from service prior to service completion.

        The one result that will be presented here for the preemptive case once again fits the model of Figure 4.15 with a single server. However, it is now assumed that the service time distribution is negative exponential and, in addition, that all classes of users have the same expected service rate . As we have noted already, the preemptive resume and the preemptive repeat cases are now identical because of the lack of memory of the server.
        For this model it can be shown (see, e.g., [KLEI 76]) that

   (4.113)

where _k, as usual, signifies the total expected system time¹³ for a user in class k and, as in the previous section, a_k = ₁ + ₂ + ... + _k. More results for preemptive priority systems are derived in Problem 4.9.

Example 2: Repair Work with Priorities

Consider a repair crew charged with performing work for vehicles of the local urban transit authority. Vehicles are separated into two types and break downs occur in a Poisson manner at rates of ₁, and ₂ for the two types of vehicles, respectively. Repair times are negative exponential and the average service time is the same, 1/, for both types. Assuming that
₁ + ₂ < , we wish to compare the expected system occupancy time (time spent waiting for repair plus time under repair) for each type of vehicle for:

      1.    The case where type 1 vehicles enjoy preemptive priority over type 2 vehicles.

      2.    The case where type 1 vehicles enjoy nonpreemptive priority over type 2 vehicles.

      3.    The case where no priorities exist and breakdowns are repaired in a FCFS fashion, irrespective of the type of vehicle.

Solution

Furthermore,

That is, the average system occupancy time for all breakdowns is identical for the three cases. The relationship shown in (4.115) is a simple example of what are often referred to in queueing theory as "conservation relations." A more general result along these lines and some interesting related questions are developed in [KLEI 76].

13 Note that (4.107) and (4.111) refer to expected waiting time.