Discrete-time stochastic processes

# A state i is transient if there is some j that is

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Unformatted text preview: that all computers are in the repair state at time 0. 3.9. EXERCISES 135 a) For a single computer, say the ith, do the epochs at which the computer enters the repair state form a renewal process? If so, ﬁnd the expected inter-renewal interval. b) Do the epochs at which it enters the operational state form a renewal process? c) Find the fraction of time over which the ith computer is operational and explain what you mean by fraction of time. d) Let Qi (t) be the probability that the ith computer is operational at time t and ﬁnd limt→1 Qi (t). e) The system is in failure mode at a given time if all computers are in the repair state at that time. Do the epochs at which system failure modes begin form a renewal process? f ) Let Pr {t} be the probability that the the system is in failure mode at time t. Find limt→1 Pr {t}. Hint: look at part (d). g) For δ small, ﬁnd the probability that the system enters failure mode in the interval (t, t + δ ] in the limit as t → 1. h) Find the expected time between successive entries into failure mode. i) Next assume that the repair time of each computer has an arbitrary density rather than exponential, but has a mean repair time of 1/∏. Do the epochs at which system failure modes begin form a renewal process? j) Repeat part (f ) for the assumption in (i). Exercise 3.27. Let {N1 (t); t≥0} and {N2 (t); t≥0} be independent renewal counting processes. Assume that each has the same distribution function F (x) for interarrival intervals and assume that a density f (x) exists for the interarrival intervals. a) Is the counting process {N1 (t) + N2 (t); t ≥ 0} a renewal counting process? Explain. b) Let Y (t) be the interval from t until the ﬁrst arrival (from either process) after t. Find an expression for the distribution function of Y (t) in the limit t → 1 (you may assume that time averages and ensemble-averages are the same). c) Assume that a reward R of rate 1 unit per second starts to be earned whenever an arrival from process 1 occurs and ceases to be earned whenever an arrival from process 2 occurs. Rt Assume that limt→1 (1/t) 0 R(τ ) dτ exists with probability 1 and ﬁnd its numerical value. d) Let Z (t) be the interval from t until the ﬁrst time after t that R(t) (as in part c) changes value. Find an expression for E [Z (t)] in the limit t → 1. Hint: Make sure you understand why Z (t) is not the same as Y (t) in part b). You might ﬁnd it easiest to ﬁrst ﬁnd the expectation of Z (t) conditional on both the duration of the {N1 (t); t ≥ 0} interarrival interval containing t and the duration of the {N2 (t); t ≥ 0} interarrival interval containing t; draw pictures! Exercise 3.28. This problem provides another way of treating ensemble-averages for renewalreward problems. Assume for notational simplicity that X is a continuous random variable. 136 CHAPTER 3. RENEWAL PROCESSES a) Show that Pr {one or more arrivals in (τ , τ + δ )} = m(τ + δ ) − m(τ ) − o(δ ) where o(δ ) ≥ 0 and limδ→0 o(δ )/δ = 0. b) Show that Pr {Z (t) ∈ [z , z + δ ), X (t) ∈ (x, x + δ )} is equal to [m(t − z ) − m(t − z − δ ) − o(δ )][FX (x + δ ) − FX (x)] for x ≥ z + δ . c) Assuming that m0 (τ ) = dm(τ )/dτ exists for all τ , show that the joint density of Z (t), X (t) is fZ (t),X (t) (z , x) = m0 (t − z )fX (x) for x &gt; z . Rt R1 d) Show that E [R(t)] = z=0 x=z R(z , x)fX (x)dx m0 (t − z )dz Rt R1 Note: Without densities, this becomes z=0 x=z R(z , x)dFX (x) dm(t − z ). This is the same R as (3.50), and if r(z ) = x≥z R(z , x)dF (x) is directly Riemann integrable, then, as shown in (3.51) to (3.53), this leads to (3.54). Exercise 3.29. This problem is designed to give you an alternate way of looking at ensemble© ™ averages for renewal-reward problems. First we ﬁnd an exact expression for Pr SN (t) &gt; s . We ﬁnd this for arbitrary s and t, 0 &lt; s &lt; t. a) By breaking the event {SN (t) &gt; s}into subevents {SN (t) &gt; s, N (t) = n}, explain each of the following steps: 1 X © ™ Pr SN (t) &gt; s = Pr {t ≥ Sn &gt; s, Sn+1 &gt; t} = = n=1 1 X n=1 Zt Z y =s = Z t y =s Pr {Sn+1 &gt;t | Sn =y } dFSn (y ) [1 − FX (t−y )] d t y =s 1 X FSn (y ) n=1 [1 − FX (t−y )] dm(y ) where m(y ) = E [N (y )] . b) Show that for 0 &lt; s &lt; t &lt; u, © ™ Pr SN (t) &gt; s, SN (t)+1 &gt; u = Z t y =s [1 − FX (u − y )] dm(y ). c) Draw a two dimensional sketch, with age and duration as the axes, and show the region of (age, duration) values corresponding to the event {SN (t) &gt; s, SN (t)+1 &gt; u}. d) Assume that for large t, dm(y ) can be approximated (according to Blackwell) as (1/X )dy , where X = E [X ]. Assuming that X also has a density, use the result in parts b) and c) to ﬁnd the joint density of age and duration. Exercise 3.30. In this problem, we show how to calculate the residual life distribution Y (t) as a transient in t. Let µ(t) = dm(t)/dt where m(t) = E [N (t)], and let the interarrival distribution have the density fX (x). Let Y...
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