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t
distribution, ﬁnd limt→1 E [R(t)]. Interpret what these quantities mean.
d) Use part (c) to ﬁnd the timeaverage expected wait per customer.
e) Find the fraction of time that there are no customers at the bus stop. (Hint: this part
is independent of a), b), and c); check your answer for E [X ] ø 1/∏).
Exercise 3.22. Consider the same setup as in Exercise 3.21 except that now customers
arrive according to a nonarithmetic renewal process independent of the bus arrival process.
Let 1/∏ be the expected interrenewal interval for the customer renewal process. Assume
that both renewal processes are in steadystate (i.e., either we look only at t ¿ 0, or we
assume that they are equilibrium processes). Given that the nth customer arrives at time
t, ﬁnd the expected wait for customer n. Find the expected wait for customer n without
conditioning on the arrival time.
Exercise 3.23. Let {N1 (t); t ≥ 0} be a Poisson counting process of rate ∏. Assume that the
arrivals from this process are switched on and oﬀ by arrivals from a nonarithmetic renewal
counting process {N2 (t); t ≥ 0} (see ﬁgure below). The two processes are independent.
rate ∏
rate ∞ ❆
✁
❆
✁
✛ On ✲
❆
✁ ❆
✁ ❆
✁ ❆
✁
❆
✁
❆
✁
✛ On ✲
❆
✁ ❆❆
✁✁ ❆
✁
❆
✁
✛ ❆
✁ ❆
✁ N2 (t)
✲ On
❆
✁ N1 (t) ❆
✁ NA (t) Let {NA (t); t ≥ 0} be the switched process; that is NA (t) includes arrivals from {N1 (t); t ≥
0} while N2 (t) is even and excludes arrivals from {N1 (t); t ≥ 0} while N2 (t) is odd.
a) Is NA (t) a renewal counting process? Explain your answer and if you are not sure, look
at several examples for N2 (t).
b) Find limt→1 1 NA (t) and explain why the limit exists with probability 1. Hint: Use
t
symmetry—that is, look at N1 (t) − NA (t). To show why the limit exists, use the renewalreward theorem. What is the appropriate renewal process to use here?
c) Now suppose that {N1 (t); t≥0} is a nonarithmetic renewal counting process but not a
Poisson process and let the expected interrenewal interval be 1/∏. For any given δ , ﬁnd
limt→1 E [NA (t + δ ) − NA (t)] and explain your reasoning. Why does your argument in (b)
fail to demonstrate a timeaverage for this case? 134 CHAPTER 3. RENEWAL PROCESSES Exercise 3.24. An M/G/1 queue has arrivals at rate ∏ and a service time distribution
given by FY (y ). Assume that ∏ < 1/E [Y ]. Epochs at which the system becomes empty
deﬁne a renewal process. Let FZ (z ) be the distribution of the interrenewal intervals and
let E [Z ] be the mean interrenewal interval.
a) Find the fraction of time that the system is empty as a function of ∏ and E [Z ]. State
carefully what you mean by such a fraction.
b) Apply Little’s theorem, not to the system as a whole, but to the number of customers
in the server (i.e., 0 or 1). Use this to ﬁnd the fraction of time that the server is busy.
c) Combine your results in a) and b) to ﬁnd E [Z ] in terms of ∏ and E [Y ]; give the fraction
of time that the system is idle in terms of ∏ and E [Y ].
d) Find the expected duration of a busy period.
Exercise 3.25. Consider a sequence X1 , X2 , . . . of IID binary random variables. Let p and
1 − p denote Pr {Xm = 1} and Pr {Xm = 0} respectively. A renewal is said to occur at time
m if Xm−1 = 0 and Xm = 1.
a) Show that {N (m); m ≥ 0} is a renewal counting process where N (m) is the number of
renewals up to and including time m and N (0) and N (1) are taken to be 0.
b) What is the probability that a renewal occurs at time m, m ≥ 2 ?
c) Find the expected interrenewal interval; use Blackwell’s theorem here.
d) Now change the deﬁnition of renewal; a renewal now occurs at time m if Xm−1 = 1 and
∗
∗
Xm = 1. Show that {Nm ; m ≥ 0} is a delayed renewal counting process where Nm is the
∗ = N ∗ = 0).
number of renewals up to and including m for this new deﬁnition of renewal (N0
1
e) Find the expected interrenewal interval for the renewals of part d).
f ) Given that a renewal (according to the deﬁnition in (d)) occurs at time m, ﬁnd the
expected time until the next renewal, conditional, ﬁrst, on Xm+1 = 1 and, next, on Xm+1 =
0. Hint: use the result in e) plus the result for Xm+1 = 1 for the conditioning on Xm+1 = 0.
g) Use your result in f ) to ﬁnd the expected interval from time 0 to the ﬁrst renewal
according to the renewal deﬁnition in d).
h) Which pattern requires a larger expected time to occur: 0011 or 0101
i) What is the expected time until the ﬁrst occurrence of 0111111?
Exercise 3.26. A large system is controlled by n identical computers. Each computer
independently alternates between an operational state and a repair state. The duration
of the operational state, from completion of one repair until the next need for repair, is
a random variable X with ﬁnite expected duration E [X ]. The time required to repair a
computer is an exponentially distributed random variable with density ∏e−∏t . All operating
durations and repair durations are independent. Assume...
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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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