Queues - Theory and Applications Problem Set 3 | Massachusetts Institute of Technology - Edubirdie

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

2023
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155

Pages:

1
Author:

rivalizesnc7

Homework Assignment 3 Problem 1 Give an example of a queueing system and performance measure such that con servation law holds for admissible scheduling policies, but does not hold if the scheduler knows the service times of the jobs in the queue in advance (and thus the policy is not admissible). HINT: work with queueing system with just one class. Problem 2 Suppose X is a random variable distributed according to distribution function F (x) and belongs to γMRLA class. Namely � ∞ P(X > τ )dτ ≤ γP(X > t) (1) t for every t ≥ 0. A. Show that E[X 2 ] ≤ γ. 2E[X] HINT: integrate both parts of (1). You may assume that E[X 2 ] < ∞. B. Suppose that interarrival times An in G/G/1 system belong to γMRLA class. Argue from this that the idle period I in a G/G/1 queueing system satisﬁes E[I 2 ] ≤ γ. 2E[I] You may assume that I has ﬁnite second moment: E[I 2 ] < ∞. Problem 3 Establish the following identity corresponding to the Region IV for G/M/m queueing system. Let j < m < i + 1. Then � ∞� � � � m −jµt � t (mµy)i−m −µy Pij = (e − e−µt )m−j mµdy dA(t). e j (i − m)! 0 0 HINT. As in other cases condition on the duration of the interarrival time.

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