From Algorithms to Z-Scores - matloff - University of California, Davis

21.2. M/M/1 431
• Due to the memoryless property, S1,resid has the same distribution as do the other Si.
• The distribution of the service times Si and queue length N observed by our tagged job is the
same as the distributions of those quantities at all times, not just at arrival times of tagged
jobs. This property can be proven for this kind of queue and many others, and is called
PASTA—Poisson Arrivals See Time Averages.
(Note that the PASTA property is not obvious. On the contrary, given our experience with
the Bus Paradox and length-biased sampling in Section 18.2, we should be wary of such
things. But the PASTA property does hold and can be proven.)
But that last term in (21.14), E[g(w) N+1 ], is the generating function of N+1, evaluated at g(w).
And we know from Section 21.2.2 that N+1 has a geometric distribution. The generating function
for a nonnegative-integer valued random variable K with success probability p is
gK(s) = E(s K ) =
∞
s i (1 − p) i−1 p =
i=1
In (21.14), we have p = 1 -u and s = g(w). So,
E(v N+1 ) =
Finally, by definition of Laplace transform,
g(w) = E(e −wS ∞
) =
0
g(w)(1 − u)
1 − u[g(w)]
ps
1 − s(1 − p)
e −wt µe −µt dt = µ
w + µ
So, from (21.11), (21.17) and (21.18), the Laplace transform of R is
µ(1 − u)
w + µ(1 − u)
(21.16)
(21.17)
(21.18)
(21.19)
In principle, Laplace transforms can be inverted, and we could use numerical methods to retrieve
the distribution of R from (21.19). But hey, look at that! Equation (21.19) has the same form as
(21.18). In other words, we have discovered that R has an exponential distribution too, only with
parameter µ(1 − u) instead of µ.
This is quite remarkable. The fact that the service and interarrival times are exponential doesn’t
mean that everything else will be exponential too, so it is surprising that R does turn out to have
an exponential distribution.