4.16 M/Hk/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 Cx 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/Hk/1 (or M/Hk/m) models could be applicable.