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Unformatted text preview: e of a customer conditional on the service time X
of that customer being 1. Find (in terms of C ) the expected system time of a customer
conditional on X = 2; (Hint: compare a customer with X = 2 to two customers with X = 1
each); repeat for arbitrary X = x.
h) Find the constant C . Hint: use f ) and g); don’t do any tedious calculations.
Exercise 6.19. Consider a queueing system with two classes of customers, type A and type
B . Type A customers arrive according to a Poisson process of rate ∏A and customers of type
B arrive according to an independent Poisson process of rate ∏B . The queue has a FCFS
server with exponentially distributed IID service times of rate µ > ∏A + ∏B . Characterize
the departure process of class A customers; explain carefully. Hint: Consider the combined
arrival process and be judicious about how to select between A and B types of customers.
Exercise 6.20. Consider a preemptive resume last come ﬁrst serve (LCFS) queueing system with two classes of customers. Type A customer arrivals are Poisson with rate ∏A and
Type B customer arrivals are Poisson with rate ∏B . The service time for type A customers
is exponential with rate µA and that for type B is exponential with rate µB . Each service
time is independent of all other service times and of all arrival epochs. 6.9. EXERCISES 275 Deﬁne the “state” of the system at time t by the string of customer types in the system at
t, in order of arrival. Thus state AB means that the system contains two customers, one
of type A and the other of type B ; the type B customer arrived later, so is in service. The
set of possible states arising from transitions out of AB is as follows:
AB A if another type A arrives.
AB B if another type B arrives.
A if the customer in service (B ) departs.
Note that whenever a customer completes service, the next most recently arrived resumes
service, so the state changes by eliminating the ﬁnal element in the string.
a) Draw a graph for the states of the process, showing all states with 2 or fewer customers
and a couple of states with 3 customers (label the empty state as E ). Draw an arrow,
labelled by the rate, for each state transition. Explain why these are states in a Markov
process.
b) Is this process reversible. Explain. Assume positive recurrence. Hint: If there is a
transition from one state S to another state S 0 , how is the number of transitions from S to
S 0 related to the number from S 0 to S ?
c) Characterize the process of type A departures from the system (i.e., are they Poisson?;
do they form a renewal process?; at what rate do they depart?; etc.)
d) Express the steady state probability Pr {A} of state A in terms of the probability of
the empty state Pr {E }. Find the probability Pr {AB } and the probability Pr {AB B A} in
terms of Pr {E }. Use the notation ρA = ∏A /µA and ρB = ∏B /µB .
e) Let Qn be the probability of n customers in the system, as a function of Q0 = Pr {E }.
Show that Qn = (1 − ρ)ρn where ρ = ρA + ρB .
Exercise 6.21. a) Generalize Exercise 6.20 to the case in which there are m types of
customers, each with independent Poisson arrivals and each with independent exponential
service times. Let ∏i and µi be the arrival rate and service rate respectively of the ith user.
Let ρi = ∏i /µi and assume that ρ = ρ1 + ρ2 + · · · + ρm < 1. In particular, show, as before
that the probability of n customers in the system is Qn = (1 − ρ)ρn for 0 ≤ n < 1.
b) View the customers in part a) as Psingle type of customer with Poisson arrivals of rate
a
P
∏ = i ∏i and with a service density i (∏i /∏)µi exp(−µi x). Show that the expected service
time is ρ/∏. Note that you have shown that, if a service distribution can be represented as
a weighted sum of exponentials, then the distribution of customers in the system for LCFS
service is the same as for the M/M/1 queue with equal mean service time. Exercise 6.22. Consider a k node Jackson type network with the modiﬁcation that each
node i has s servers rather than one server. Each server at i has an exponentially distributed
service time with rate µi . The exogenous input rate to node i is ρi = ∏0 Q0i and each output
from i is switched to j with probability Qij and switched out of the system with probability 276 CHAPTER 6. MARKOV PROCESSES WITH COUNTABLE STATE SPACES Qi0 (as in the text). Let ∏i , 1 ≤ i ≤ k, be the solution, for given ∏0 , to
∏j = k
X ∏i Qij ; i=0 1 ≤ j ≤ k and assume that ∏i < sµi ; 1 ≤ i ≤ k. Show that the steady state probability of
state m is
Pr {m } = k
Y pi (mi ), i=1 where pi (mi ) is the probability of state mi in an (M , M , s) queue. Hint: simply extend the
argument in the text to the multiple server case.
Exercise 6.23. Suppose a Markov process with the set of states A is reversible and has
steady state probabilities pi ; i ∈ A. Let B be a subset of A and assume that the process
is changed by setting qij = 0 for all i ∈ B , j ∈ B . Assuming that the new process (starting
/
in B and remining in...
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This note was uploaded on 09/27/2010 for the course EE 229 taught by Professor R.srikant during the Spring '09 term at University of Illinois, Urbana Champaign.
 Spring '09
 R.Srikant

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