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the system is given by Little’s law again as L = W λ = Lq + λ/µ. Thus, in terms of Lq ,
L = Lq + λ/µ
W = Lq /λ + 1/µ
Wq = Lq /λ M/M/∞: There are no customers waiting for service, so Lq = Wq = 0. Each customer
waits in the system for its own service time, so W = 1/µ. By Little’s formula, L = λ/µ.
Exercise 6.1:
a) The holding interval U1 conditional on X0 = i is exponentially distributed with
parameter vi . And vi is uniquely determined by transition rates qij as:
�
qij = qi,i+1 + qi,i−1 = λ + µ
vi =
j Thus, E[U1 X0 = i] = 1/vi = 1/(λ + µ).
b) The holding interval Un between the time that state Xn−1 = l is entered and Xn
entered, conditional on Xn−1 is jointly independent of Xm for all m = n − 1. So, E[U1 X0 =
�
i, X1 = i + 1] = E[U1 X0 = i] = 1/vi = 1/(λ + µ).
The same is true for E[U1 X0 = i, X1 = i + 1].
c) Conditional on {X0 = i, Xi+1 = i + 1}, we know that the ﬁrst transition is an arrival,
so the ﬁrst arrival time (V ) is the same a...
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This document was uploaded on 03/19/2014 for the course EECS 6.262 at MIT.
 Spring '11
 RobertGallager

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