5.5 Generating a Nonhomogeneous Poisson Process
An extremely important counting process for modeling purposes is the nonhomogeneous Poisson process, which relaxes the Poisson process assumption of stationary increments. Thus, it allows for the possibility that the arrival rate need not be constant but can vary with time. It is usually very difficult to obtain analytical results for a mathematical model that assumes a nonhomogeneous Poisson arrival process, and as a result such processes are not applied as often as they should be.
However, because simulation can be used to analyze such models, we expect that such mathematical models will become more common.
Suppose that we wanted to simulate the firstT time units of a nonhomogeneous Poisson process with intensity functionλ(t). The first method we present, called thethinningorrandom samplingapproach, starts by choosing a valueλwhich is such that
λ(t)⩽λ for allt ⩽T
Now, as shown in Chapter 2, such a nonhomogeneous Poisson process can be generated by a random selection of the event times of a Poisson process having rateλ. That is, if an event of a Poisson process with rateλthat occurs at timet is counted (independently of what has transpired previously) with probabilityλ(t)/λ, then the process of counted events is a nonhomogeneous Poisson process with intensity functionλ(t),0⩽t ⩽T. Hence, by simulating a Poisson process and then randomly counting its events, we can generate the desired nonhomogeneous Poisson process. This can be written algorithmically as follows.
Generating the First T Time Units of a Nonhomogeneous Poisson Process
step 1: t=0,I =0.
step 2: Generate a random numberU.
step 3: t=t−1λlogU. Ift >T, stop.
step 4: Generate a random numberU.
step 5: IfU ⩽λ(t)/λ, setI =I+1,S(I)=t. step 6: Go to Step 2.
In the aboveλ(t)is the intensity function andλis such thatλ(t)⩽λ. The final value of I represents the number of events time T, andS(1), . . . ,S(I)are the event times.
The above procedure, referred to as the thinning algorithm (because it “thins”
the homogeneous Poisson points), is clearly most efficient, in the sense of having the fewest number of rejected events times, when λ(t)is nearλthroughout the interval. Thus, an obvious improvement is to break up the interval into subintervals and then use the procedure over each subinterval. That is, determine appropriate valuesk,0=t0<t1<t2<ã ã ã<tk <tk+1=T, λ1, . . . , λk+1such that
λ(s)⩽λi ifti−1⩽s<ti, i =1, . . . ,k+1 (5.7)
Now generate the nonhomogeneous Poisson process over the interval(ti−1,ti)by generating exponential random variables with rateλi, and accepting the generated event occurring at times,s ∈ (ti−1,ti), with probabilityλ(s)/λi. Because of the memoryless property of the exponential and the fact that the rate of an exponential can be changed upon multiplication by a constant, it follows that there is no loss of efficiency in going from one subinterval to the next. That is, if we are att∈(ti−1,ti) and generate X, an exponential with rateλi, which is such thatt+X >ti, then we can useλi[X−(ti−t)]/λi+1as the next exponential with rateλi+1.
We thus have the following algorithm for generating the firstT time units of a nonhomogeneous Poisson process with intensity functionλ(s)when the relations (5.7) are satisfied. In the algorithm t represents the present time, J the present interval (i.e., J = j whentj−1 ⩽ t < tj), I the number of events so far, and S(1), . . . ,S(I)the event times.
Generating the First T Time Units of a Nonhomogeneous Poisson Process
step 1: t =0,J=1,I =0.
step 2: Generate a random numberU and setX = −λJ1logU.
step 3: Ift+X>tJ, go to Step 8.
step 4: t =t+X.
step 5: Generate a random numberU.
step 6: IfU ⩽λ(t)/λJ, setI =I+1,S(I)=t. step 7: Go to Step 2.
step 8: IfJ =k+1, stop.
step 9: X =(X−tJ +t)λJ/λJ+1,t =tJ,J =J+1.
step 10: Go to Step 3.
Suppose now that over some subinterval(ti−1,ti)we have thatλi>0, where λi ≡Infimum{λ(s):ti−1⩽s<ti}
In such a situation we should not use the thinning algorithm directly but rather should first simulate a Poisson process with rateλi over the desired interval and then simulate a nonhomogeneous Poisson process with the intensity function λ(s) = λ(s)−λi when s ∈ (ti−1,ti). (The final exponential generated for the Poisson process, which carries one beyond the desired boundary, need not be wasted but can be suitably transformed so as to be reusable.) The superposition (or merging) of the two processes yields the desired process over the interval. The reason for doing it this way is that it saves the need to generate uniform random variables for a Poisson distributed number, with meanλi(ti−ti−1), of the event times. For example, consider the case where
λ(s)=10+s, 0<s<1
5.5 Generating a Nonhomogeneous Poisson Process 87 Using the thinning method withλ =11 would generate an expected number of 11 events, each of which would require a random number to determine whether or not it should be accepted. On the other hand, to generate a Poisson process with rate 10 and then merge it with a nonhomogeneous Poisson process with rate λ(s)= s,0 <s <1 (generated by the thinning algorithm withλ =1), would yield an equally distributed number of event times but with the expected number needing to be checked to determine acceptance being equal to 1.
A second method for simulating a nonhomogeneous Poisson process having intensity functionλ(t),t>0, is to directly generate the successive event times. So letS1,S2, . . .denote the successive event times of such a process. As these random variables are clearly dependent, we generate them in sequence—starting withS1, and then using the generated value ofS1to generateS2, and so on.
To start, note that if an event occurs at times, then, independent of what has occurred prior tos, the additional time until the next event has the distributionFs, given by
Fs(x)= P{time fromsuntil next event is less thanx|event ats}
= P{next event is beforex+s|event ats}
= P{event betweensands+x|event ats}
= P{event betweensands+x}by independent increments
=1−P{0 events in(s,s+x)}
=1−exp
− s+x
s
λ(y)d y
=1−exp
− x
0
λ(s+y)d y
(5.8) We can now simulate the event times S1,S2, . . . by generating S1 from the distributionF0; if the simulated value ofS1iss1, we generateS2by addings1to a generated value from the distribution Fs1; if this sum iss2 we generate S3 by addings2to a generated value from the distribution Fs2; and so on. The method used to simulate from these distributions should of course depend on their form.
In the following example the distributionsFsare easily inverted and so the inverse transform method can be applied.
Example 5h Suppose thatλ(t)=1/(t+a),t ⩾0, for some positive constant a. Then
x 0
λ(s+y)d y= x
0
1
s+y+ad y=log
x+s+a s+a
Hence, from Equation (5.8),
Fs(x)=1− s+a
x+s+a = x x+s+a
To invert this, suppose thatx=Fs−1(u), and so u =Fs(x)= x
x+s+a or, equivalently,
x= u(s+a) 1−u That is,
Fs−1(u)=(s+a) u 1−u
We can therefore generate the successive event times S1,S2, . . .by generating random numbersU1,U2, . . .and then recursively setting
S1= aU1 1−U1
S2=S1+(S1+a) U2 1−U2
= S1+aU2 1−U2
and, in general,
Sj =Sj−1+(Sj−1+a) Uj
1−Uj
= Sj−1+aUj
1−Uj
, j ⩾2