In some systems, a customer is allocated to one of two service facilities. If the servic...

In some systems, a customer is allocated to one of two service facilities. If the service time for a customer served by facility i has an exponential distribution with parameter λ_i (i = 1, 2) and p is the proportion of all customers served by facility 1, then the pdf of X = the service time of a randomly selected customer is

This is often called the hyperexponential or mixed exponential distribution. This distribution is also proposed as a model for ainfall amount in “Modeling Monsoon Affected Rainfall of Pakistan by Point Processes” (J. of Water Resources Planning and Mgmnt., 1992: 671–688).

a. Verify that f(x; λ₁, λ₂, p)is indeed a pdf.

b. What is the cdf ?

c. If X has f(x; λ₁, λ₂, p) as its pdf, what is E(X)?

d. Using the fact that E(X² )= 2/λ² when X has an exponential distribution with parameter λ , compute E(X² ) when X has pdf f(x; λ₁, λ₂, p) . Then compute V(X).

e. The coefficient of variation of a random variable (or distribution) is . What is CV =σ/μ for an exponential rv? What can you say about the value of CV when X has a hyperexponential distribution?

f. What is CV for an Erlang distribution with parameters and n as defined in Exercise 68? [Note: In applied work, the sample CV is used to decide which of the three distributions might be appropriate.]

Reference exercise 68

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

p>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)