University:
Massachusetts Institute of Technology
Course:
15.501/516 | Accounting
Academic year:

2023
Views:

162

Pages:

2
Author:

enecemaallolo8qo

Midterm Exam Problem 1 For the following questions/statements just give TRUE or FALSE answers. Do not derive the answers. Consider a G/G/1 queueing system. The arrival rate is λ and service rate is µ > λ. A. Let L10 be the steady state number of customers in positions 1 to 10 (customer in service is assumed to be in position 1). Let S10 be the steady state time that a typical customer was in one of the positions 110. Then the Little’s Law holds, namely E[L10 ] = λE[S10 ]. B. The distributional law holds for L10 and S10 when the scheduling policy is (i) FirstInFirstOut (ii) LastInFirstOut C. Suppose that the system has instead two servers (that is we have G/G/2 queueing λ system). Then the probability that the system is empty is 1 − ρ, where ρ = 2µ . Problem 2 Consider an M/G/1 queueing system where arrival rate λ = 1 and service time with a mixed distribution. Namely, with probability 1/2 it is exponential with rate 2 and with probability 1/2 it is exponential with rate 3. Assume the system operates under the FirstInFirstOut policy. A. Compute the traﬃc intensity ρ of this system. B. Suppose the revenue obtained from a customer is e−cy if the customer waited y time units. Compute the expected steady state revenue when c = 1. C. Extra credit. Suppose for every customer who waited y time units the cost ecy is paid. What is the largest c0 for which the expected cost is ﬁnite? Is c0 < 1.9? Problem 3 M/D/1 Queueing system with feedback. Namely each served customer comes back to the queue with some probability p and leaves the system with probability 1 − p. The service time is assumed to be deterministic with value d both for the initial and returning customers. The arrival process is Poisson with rate λ. 2 Midterm Exam • Under which conditions on λ, d, p is there a steadystate regime? Do not prove this, just provide a right answer. • Compute the expected number of customers in the queue. HINT: Observe that the number in the system is policy invariant for all nonpreemptive work conserving scheduling policies. Useful facts The expected waiting time in M/G/1 queueing system under FirstComeFirst Serve policy is E[W ] = λE[X 2 ] , 2(1 − ρ) and the Laplace Transform of the waiting time is φW (s) = (1 − ρ)s , λβ(s) − λ + s where λ is the arrival rate, X is service time, ρ = λE[X] and β(s) is the Laplace Transform of the service time distribution.