The special case of the gamma distribution in which α is a positive integer n is called an Erlang distribution. If we replace β by
1/λin Expression (4.8), the Erlang pdf is
It can be shown that if the times between successive events are independent, each with an exponential distribution with parameter , then the total time X that elapses before all of the next n events occur has pdf f(x; λ ,n).
a. What is the expected value of X ? If the time (in minutes) between arrivals of successive customers is exponentially distributed with λ = .5, how much time can be expected to elapse before the tenth customer arrives?
b. >If customer interarrival time is exponentially distributed with λ =.5 , what is the probability that the tenth customer (after the one who has just arrived) will arrive within the next 30 min?
c. The event { X ≤ 5} occurs iff at least n events occur in the next t units of time. Use the fact that the number of events occurring in an interval of length t has a Poisson distribution with parameter lt to write an expression(involving Poisson probabilities) for the Erlang cdf F(t; λ, n)= P(X ≤ t)
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