Three types of jobs arrive at a computer server: With probability p1, a type 1 job arrives, requiring an exponentially distributed service time T1 with parameter , i.e., fT1 (t1) = et1 ,

With probability p2, a type 2 job arrives, requiring a uniformly distributed service time T2 in the interval [0, T],

With probability p3 = 1 p1 p2, a type 3 job arrives, requiring a fixed service time T3 = K, where K < T.

(a) Find P(T1 K).

(b) Find P(T2 K).

(c) Find the probability that service time of an arbitrary job, arriving according to the distribution (p1, p2, p3), is less than or equal to K.

(d) If service time of a job is known to be less than or equal to K, what is the probability that it is a type 1 job?