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Unformatted text preview: w that Pi ≤ P1 [FX (δ )]i−1 . c) Show that E [N (t + δ ) − N (t)] ≤ P1 [1 − FX (δ )]−2 .
d) Show that E [N (t + δ ) − N (t)] ≤ P1 [1 − o(δ )].
e) Show that E [N (t + δ ) − N (t)] ≥ P1 .
f ) Use parts d) and e), along with Blackwell’s theorem, to verify (3.45) and the three
equations of (3.19).
Exercise 3.16. Let Y (t) = SN (t)+1 − t be the residual life at time t of a renewal process.
First consider a renewal process in which the interarrival time has density fX (x) = e−x ; x ≥
0, and next consider a renewal process with density
fX (x) = 3
;
(x + 1)4 x ≥ 0. For each of the above densities, use renewalreward theory to ﬁnd:
i) the timeaverage of Y (t)
ii) the second moment in time of Y (t) (i.e., limT →1 1
T RT
0 Y 2 (t)dt) £
§
For the exponential density, verify your answers by ﬁnding E [Y (t)] and E Y 2 (t) directly. Exercise 3.17. Consider a variation of an M/G/1 queueing system in which there is no
facility to save waiting customers. Assume customers arrive according to a Poisson process
of rate ∏. If the server is busy, the customer departs and is lost forever; if the server is
not busy, the customer enters service with a service time distribution function denoted by
FY (y ).
Successive service times (for those customers that are served) are IID and independent of
arrival times. Assume that customer number 0 arrives and enters service at time t = 0.
a) Show that the sequence of times S1 , S2 , . . . at which successive customers enter service are
the renewal times of a renewal process. Show that each interrenewal interval Xi = Si − Si−1
(where S0 = 0) is the sum of two independent random variables, Yi + Ui where Yi is the ith
service time; ﬁnd the probability density of Ui .
b) Assume that a reward (actually a cost in this case) of one unit is incurred for each
customer turned away. Sketch the expected reward function as a function of time for the
sample function of interrenewal intervals and service intervals shown below; the expectation
is to be taken over those (unshown) arrivals of customers that must be turned away.
Rt
c) Let 0 R(τ )dτ denote the accumulated reward (i.e., cost) from 0 to t and ﬁnd the limit
Rt
as t → 1 of (1/t) 0 R(τ )dτ . Explain (without any attempt to be rigorous or formal) why
this limit exists with probability 1. 132 CHAPTER 3. RENEWAL PROCESSES ✛
❄ Y1 ✲ S0 = 0 ❄ ✛
❄ Y1 S1 ✲
❄ ✛
❄ S2 Y1 ✲
❄ d) In the limit of large t, ﬁnd the expected reward from time t until the next renewal. Hint:
Sketch this expected reward as a function of t for a given sample of interrenewal intervals
and service intervals; then ﬁnd the timeaverage.
e) Now assume that the arrivals are deterministic, with the ﬁrst arrival at time 0 and
the nth arrival at time n − 1. Does the sequence of times S1 , S2 , . . . at which subsequent
customers start service still constitute the renewal times of a renewal process? Draw a sketch
≥R
¥
t
of arrivals, departures, and service time intervals. Again ﬁnd limt→1 0 R(τ ) dτ /t.
Exercise 3.18. Let Z (t) = t − SN (t) be the age of a renewal process and Y (t) = SN (t)+1 − t
be the residual life. Let FX (x) be the distribution function of the interrenewal interval and
ﬁnd the following as a function of FX (x):
a) Pr {Y (t)>x  Z (t)=s}
b) Pr {Y (t)>x  Z (t+x/2)=s}
c) Pr {Y (t)>x  Z (t+x)>s} for a Poisson process.
Exercise 3.19. Let Z (t), Y (t), X (t) denote the age, residual life, and duration at time t
for a renewal counting process {N (t); t ≥ 0} in which the interarrival time has a density
given by f (x). Find the following probability densities; assume steadystate.
a) fY (t) (y  Z (t+s/2)=s) for given s > 0.
b) fY (t),Z (t) (y , z ).
c) fY (t) (y  X (t)=x).
d) fZ (t) (z  Y (t−s/2)=s) for given s > 0.
e) fY (t) (y  Z (t+s/2)≥s) for given s > 0.
Exercise 3.20. a) Find limt→1 {E [N (t)] − t/X } for a renewal counting process {N (t); t ≥
0} with interrenewal times {Xi ; i ≥ 1}. Hint: use Wald’s equation.
b) Evaluate your result for the case in which X is an exponential random variable (you
already know what the result should be in this case).
£§
c) Evaluate your result for a case in which E [X ] < 1 and E X 2 = 1. Explain (very
brieﬂy) why this does not contradict the elementary renewal theorem.
Exercise 3.21. Customers arrive at a bus stop according to a Poisson process of rate ∏.
Independently, buses arrive according to a renewal process with the interrenewal interval 3.9. EXERCISES 133 distribution FX (x). At the epoch of a bus arrival, all waiting passengers enter the bus and
the bus leaves immediately. Let R(t) be the number of customers waiting at time t.
a) Draw a sketch of a sample function of R(t).
b) Given that the ﬁrst£bus arrives at time X1 = x, ﬁnd the expected number of customers
§
Rx
picked up; then ﬁnd E 0 R(t)dt , again given the ﬁrst bus arrival at X1 = x.
Rt
c) Find limt→1 1 0 R(τ )dτ (with probability 1). Assuming that FX...
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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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