Discrete-time stochastic processes

# tn 234 two proofs are given each illustrative of

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Unformatted text preview: 0, also satisﬁes: n o e Pr N (t, t + δ ) = 0 = 1 − δ ∏(t) + o(δ ) n o e Pr N (t, t + δ ) = 1 = δ ∏(t) + o(δ ) n o e Pr N (t, t + δ ) ≥ 2 = o(δ ). (2.26) e where N (t, t + δ ) = N (t + δ ) − N (t). The non-homogeneous Poisson process does not have the stationary increment property. One common application occurs in optical communication where a non-homogeneous Poisson process is often used to model the stream of photons from an optical modulator; the modulation is accomplished by varying the photon intensity ∏(t). We shall see another application shortly in the next example. Sometimes a Poisson process, as we deﬁned it earlier, is called a homogeneous Poisson process. We can use a “shrinking Bernoulli process” again to approximate a non-homogeneous Poisson process. To see how to do this, assume that ∏(t) is bounded away from zero. We partition the time axis into increments whose lengths δ vary inversely with ∏(t), thus holding the probability of an arrival in an increment at some ﬁxed value q = δ ∏(t). Thus, 8 We assume that ∏(t) is right continuous, i.e., that for each t, ∏(t) is the limit of ∏(t + ε) as ε approaches 0 from above. This allows ∏(t) to contain discontinuities, as illustrated in Figure 2.7, but follows the convention that the value of the function at the discontinuity is the limiting value from the right. This convention is required in (2.26) to talk about the distribution of arrivals just to the right of time t. 2.4. NON-HOMOGENEOUS POISSON PROCESSES temporarily ignoring the variation of ∏(t) within an increment, Ωµ ∂ æ e t, t + q Pr N =0 = 1 − q + o(q ) ∏(t) Ωµ ∂ æ e t, t + q Pr N =1 = q + o(q ) ∏(t) Ωµ ∂ æ q e Pr N t, t + ≥2 = o(ε). ∏(t) Ths partition is deﬁned more precisely by deﬁning m(t) as Zt m(t) = ∏(τ )dτ . 75 (2.27) (2.28) 0 Then the ith increment ends at that t for which m(t) = q i. r t ❅ ❅ ∏(t) ❅ ❅ Figure 2.7: Partitioning the time axis into increments each with an expected number of arrivals equal to q . Each rectangle above has the same area, which ensures that the ith partition ends where m(t) = q i. As before, let {Yi ; i ≥ 1} be a sequence of IID binary rv’s with Pr {Yi = 1} = q and Pr {Yi = 0} = 1 − q . Consider the counting process {N (t); t ≥ 0} in which Yi , for each i ≥ 1, denotes the number of arrivals in the interval (ti−1 , ti ], where ti satisﬁes m(ti ) = iq . Thus, N (ti ) = Y1 + Y2 + · · · + Yi . If q is decreased as 2−j each increment is successively split into a pair of increments. Thus by the same argument as in (2.21), [1 + o(q )][m(t)]n exp[−m(t)] . (2.29) n! Rτ Similarly, for any interval (t, τ ], taking m(t, τ ) = t ∏(u)du, and taking t = tk , τ = ti for e some k, i, we get n o [1 + o(q )][m(t, τ )]n exp[−m(t, τ )] e e e Pr N (t, τ ) = n = . (2.30) n! Pr {N (t) = n} = Going to the limit q → 0, the counting process {N (t); t ≥ 0} above approaches the nonhomogeneous Poisson process under consideration, and we have the following theorem: Theorem 2.5. For a non-homogeneous Poisson process with right-continuous arrival rate e ∏(t) bounded away from zero, the distribution of N (t, τ ), the number of arrivals in (t, τ ], satisﬁes Zτ n o [m(t, τ )]n exp[−m(t, τ )] e e e Pr N (t, τ ) = n = where m(t, τ ) = e ∏(u) du. (2.31) n! t 76 CHAPTER 2. POISSON PROCESSES Hence, one can view a non-homogeneous Poisson process as a (homogeneous) Poisson process over a non-linear time scale. That is, let {N ∗ (s); s ≥ 0} be a (homogeneous) Poisson process with rate 1. The non-homogeneous Poisson process is then given by N (t) = N ∗ (m(t)) for each t. Example 2.4.1 (THE M/G/1 Queue). Queueing theorists use a standard notation of characters separated by slashes to describe common types of queueing systems. The ﬁrst character describes the arrival process to the queue. M stands for memoryless and means a Poisson arrival process; D stands for deterministic and means that the interarrival interval is ﬁxed and non-random; G stands for general interarrival distribution. We assume that the interarrival intervals are IID (thus making the arrival process a renewal process), but many authors use GI to explicitly indicate IID interarrivals. The second character describes the service process. The same letters are used, with M indicating the exponential service time distribution. The third character gives the number of servers. It is assumed, when this notation is used, that the service times are IID, independent of the arrival times, and independent of the which server is used. With this notation, M/G/1 indicates a queue with Poisson arrivals, a general service distribution, and an inﬁnite number of servers. Similarly, the example at the end of Section 2.3 considered an M/M/1 queue. Since the M/G/1 queue has an inﬁnite number of servers, no arriving customers are ever queued. Each arrival immediately starts to be served by some server, and the service time Yi of customer i is IID over i with some distribution...
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