35. Consider the priority queuing model of Section 6.2 but now suppose that if a type 2 customer is being served when a type 1 arrives then the type 2 customer is bumped out of service. This is called the preemptive case.

Suppose that when a bumped type 2 customer goes back in service his service begins at the point where it left off when he was bumped.

(a) Argue that the work in the system at any time is the same as in the nonpreemptive case.

(b) Derive W^.

Hint: How do type 2 customers affect type Ts?

(c) Why is it not true that V^ = X2E[S2]WQ

(d) Argue that the work seen by a type 2 arrival is the same as in the nonpreemptive case, and so WQ = WQ (nonpreemptive) + Å [extra time]

where the extra time is due to the fact that he may be bumped.

(e) Let Í denote the number of times a type 2 customer is bumped.

Why is NE[SX]

Å [extra time \N] =

1 - A^ I S J Hint: When a type 2 is bumped, relate the time until he gets back in service to a "busy period."

(f) Let S2 denote the service time of a type 2. What is E[N\S2]1

(g) Combine the preceding to obtain

,2 „ , 2 , . ^ [ S . l t f i S J WQ = WQ (nonpreemptive) +

1 - ë,ÅÉÎ,]