## [數學]等候理論 (Queueing theory) ### [數學]等候理論 (Queueing theory)

1. Suppose that a queueing system has 2 servers. an exponential interarrival time distribution with a mean of 2 hours, and an exponential service-time distribution with a mean of 2 hour.
Futhermore, a customer has just arrived at 12:00 noon.

a) What is the probability that the next arrival will come before 1:00 p.m. ?Between 1:00 and 2:00 pm? After 2:oo pm?
b)Suppose that no additional customers arrive before 1:00 p.m.. Now what is the probability that the next arrival will come between 1:00 and 2:00 pm?
c)What is the probability that the numner of arrivals between 1:00 and 2:00 p.m. will be zero? One? Two or more?

2. Consider a variation of the M/M/1 model where customers renege (leavethe queueing system without being served) if their waiting time in the queue grows too large. In particular, assume that the time each customer is willing to wait in the queue before reneging has an exponential distribution with a mean of 1/x.
Construct the rate diagram for this queueing system.

yu-ju