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

 265-2.gif (4742 bytes)

       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,

formpg266-1.gif (3928 bytes)

b. Show that the coefficient of variation of X,

formpg266-2.gif (3573 bytes)

Hint: Use the Cauchy-Schawrz inequality,

formpg266-3.gif (3856 bytes)

        Because of the fact that Csquare.gif (861
bytes)x geq.gif (61 bytes) 1, hyperexponential random variables are said to be "more random" than negative exponential random variables (for which Csquare.gif (861 bytes) =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 lambda.gif (179 bytes) = 3 be the arrival rate at the system and let the service time pdf be given by

266-2.gif (2652 bytes)

        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 alpha.gif (53
bytes)1 or type alpha.gif
(53 bytes)2 service with probability alpha.gif (53
bytes)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 alpha.gif (53
bytes)1, alpha.gif (53
bytes)2, mu.gif (189
bytes)1, and mu.gif (189
bytes)2 in this case?

266-3.gif (19246 bytes)

d. Find Lbar.gif (304 bytes),Wbar.gif (557 bytes),Lbar.gif (304 bytes)q and Wbar.gif (557
bytes)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 alpha.gif (53
(179 bytes); the rate of transitions from state (2,2) to state (1,1) is equal to alpha.gif (53
bytes)1mu.gif (189

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