Discrete-time stochastic processes

1 in the graphical representation there is one node

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

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.

Ask a homework question - tutors are online