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 CauchySchawrz 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/H_{2}/1 queueing system with
infinite queue capacity
(H_{2} indicates that
service times are
secondorder 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/H_{2}/1
facility.
e. Carefully draw a statetransition diagram for
this M/H_{2}/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 _{1}_{2}.
f. Describe a possible situation in an urban
service system context
whereM/H_{k}/1 (or
M/H_{k}/m) models could
be applicable.
