Discrete-time stochastic processes

# 1 in the graphical representation there is one node

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Unformatted text preview: is a non-arithmetic t distribution, ﬁnd limt→1 E [R(t)]. Interpret what these quantities mean. d) Use part (c) to ﬁnd the time-average 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 non-arithmetic renewal process independent of the bus arrival process. Let 1/∏ be the expected inter-renewal interval for the customer renewal process. Assume that both renewal processes are in steady-state (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 non-arithmetic 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 non-arithmetic renewal counting process but not a Poisson process and let the expected inter-renewal 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 time-average 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 inter-renewal intervals and let E [Z ] be the mean inter-renewal 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 inter-renewal 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 inter-renewal 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.

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