4.16 M/H_k/m queueing systems The hyperexponential pdf of order k is the pdf

       In other words, a hyperexponential pdf can be viewed as the weighted sum of k distict negative exponential pdf's.

a. Show that, for the random variable X with hyperexponential pdf,

b. Show that the coefficient of variation of X,

Hint: Use the Cauchy-Schawrz inequality,

        Because of the fact that C_x 1, hyperexponential random variables are said to be "more random" than negative exponential random variables (for which C =1).

c. Consider now a M/H₂/1 queueing system with infinite queue capacity

(H₂ indicates that service times are second-order hyperexponential random variables.) Let = 3 be the arrival rate at the system and let the service time pdf be given by

        A schematic representation of this system is shown in Figure P4.16. Each user, upon entrance to the service facility, will receive type 1 service with probability ₁ or type ₂ service with probability ₂. Whenever either one of the two types of services is being offered, no other user can obtain access to the facility. What are the values of ₁, ₂, ₁, and ₂ in this case?

d. Find ,,_q and _q in this case. Compare with the equivalent quantities for a M/M/1 system with service rate (when busy) equal to the service rate of this M/H₂/1 facility.

e. Carefully draw a state-transition diagram for this M/H₂/1 system.

Hint 1: Define states: "0" = the system is empty; "i,j"= i users are present (i=1,2, ...) and the user currently occupying the service facility is receiving type j service    (j = 1,2).

Hint 2: The rate of transitions from state 0 to state (1,1) is equal to ₁; the rate of transitions from state (2,2) to state (1,1) is equal to ₁₂.

f. Describe a possible situation in an urban service system context whereM/H_k/1 (or M/H_k/m) models could be applicable.