Question: Consider a single-server queue with Poisson arrivals at rate λ = 10.82 per minute and normally distributed service times with mean 5.1 seconds and variance 0.98 seconds2. It is desired to estimate the mean time in the system for a customer who, upon arrival, finds i other customers in the system-that is, to estimate
wi = E(W|N = i) for i = 0, 1, 2,...
where W is a typical system time and N is the number of customers found by an arrival. For example, w0 is the mean system time for those customers who find the system empty, w1 is the mean system time for those customers who find one other customer present upon arrival, and so on. The estimate w1 of w1 will be a sample mean of system times taken over all arrivals who find i in the system. Plot w1 vs i. Hypothesize and attempt to verify a relation between w; and i.
(a) Simulate for a 10-hour period with empty and idle initial conditions.
(b) Simulate for a 1 0-hour. period after an initialization of one hour. Are there observable differences in the results of (a) and (b)?
(c) Repeat parts (a) and (b) with service times exponentially distributed with mean 5.1 seconds.
(d) Repeat parts (a) and (b) with deterministic service times equal to 5.1 seconds.
(e) Find the number of replications needed to estimate w0, w1, ...... , w6 with a standard error for each of at most 3 seconds. Repeat parts (a)-(d), but using this number of replications.