Poisson Processes

Chapter 2 POISSON PROCESSES 2.1 Introduction A Poisson process is a simple and widely used stochastic process for modeling the times at which arrivals enter a system. It is in many ways the continuous-time version of the Bernoulli process that was described in Section 1.3.5. For the Bernoulli process, the arrivals can occur only at positive integer multiples of some given increment size (often taken to be 1). Section 1.3.5 characterized the process by a sequence of IID binary random variables (rv’s), Y1 , Y2 , . . . , where Yi = 1 indicates an arrival at increment i and Yi = 0 otherwise. We observed (without any careful proof) that the process could also be characterized by the sequence of interarrival times. These interarrival times are geometrically distributed IID rv’s . For the Poisson process, arrivals may occur at arbitrary positive times, and the probability of an arrival at any particular instant is 0. This means that there is no very clean way of describing a Poisson process in terms of the probability of an arrival at any given instant. It is more convenient to define a Poisson process in terms of the sequence of interarrival times, X1 , X2 , . . . , which are defined to be IID. Before doing this, we describe arrival processes in a little more detail. 2.1.1 Arrival processes An arrival process is a sequence of increasing rv’s, 0 < S1 < S2 < · · · , where1 Si < Si+1 means that Si+1 Si is a positive rv, i.e., a rv X such that FX (0) = 0. The rv’s S1 , S2 , . . . , are called arrival epochs (the word time is somewhat overused in this subject) and represent the times at which some repeating phenomenon occurs. Note that the process starts at time 1 These rv’s Si can be viewed as sums of interarrival times. They should not be confused with the rv’s Si used in Section 1.3.5 to denote the number of arrivals by time i for the Bernoulli process. We use Si throughout to denote the sum of i rv’s. Understanding how such sums behave is a central issue of every chapter (and almost every section ) of these notes. Unfortunately, for the Bernoulli case, the IID sums of primary interest are the sums of binary rv’s at each time increment, whereas here the sums of primary interest are the sums of interarrival intervals. CHAPTER 2. POISSON PROCESSES 0 and that multiple arrivals can’t occur simultaneously (the phenomenon of bulk arrivals can be handled by the simple extension of associating a positive integer rv to each arrival). We will sometimes permit simultaneous arrivals or arrivals at time 0 as events of zero probability, but these can be ignored. In order to fully specify the process by the sequence S1 , S2 , . . . of rv’s, it is necessary to specify the joint distribution of the subsequences S1 , . . . , Sn for all n > 1. Although we refer to these processes as arrival processes, they could equally well model departures from a system, or any other sequence of incidents. Although it is quite common, especially in the simulation field, to refer to incidents or arrivals as events, we shall avoid that here. The nth arrival epoch Sn is a rv and {Sn  t}, for example, is an event. This would make it confusing to refer to the nth arrival itself as an event. X1 0 r - S1 r X2 X3 6 N (t) r - N (t) = n for Sn  n < Sn+1 t S2 S3 Figure 2.1: A sample function of an arrival process and its arrival epochs {S1 , S2 , . . . }, its interarrival intervals {X1 , X2 , . . . }, and its counting process {N (t); t > 0} As illustrated in Figure 2.1, any arrival process can also be specified by two alternative stochastic processes. The first alternative is the sequence of interarrival times, X1 , X2 , . . . ,. These are positive rv’s defined in terms of the arrival epochs by X1 = S1 and Xi = Si Si 1 for i > 1. Similarly, given the Xi , the arrival epochs Si are specified as Xn Sn = Xi . (2.1) i=1 Thus the joint distribution of X1 , . . . , Xn for all n > 1 is sufficient (in principle) to specify the arrival process. Since the interarrival times are IID in most cases of interest, it is usually much easier to specify the joint distribution of the Xi than of the Si . The second alternative for specifying an arrival process is the counting process N (t), where for each t > 0, the rv N (t) is the number of arrivals2 up to and including time t. The counting process {N (t); t > 0}, illustrated in Figure 2.1, is an uncountably infinite family of rv’s {N (t); t > 0} where N (t), for each t > 0, is the number of arrivals in the interval (0, t]. Whether the end points are included in these intervals is sometimes important, and we use parentheses to represent intervals without end points and square brackets to represent inclusion of the end point. Thus (a, b) denotes the interval {t : a < t < b}, and (a, b] denotes {t : a < t  b}. The counting rv’s N (t) for each t > 0 are then defined as the number of arrivals in the interval (0, t]. N (0) is defined to be 0 2 Thus, for the Bernoulli process with an increment size of 1, N (n) is the rv denoted as Sn in Section 1.3.5 2.2. DEFINITION AND PROPERTIES OF A POISSON PROCESS with probability 1, which means, as before, that we are considering only arrivals at strictly positive times. The counting process {N (t); t > 0} for any arrival process has the properties that N (⌧ ) N (t) for all ⌧ t > 0 (i.e., N (⌧ ) N (t) is a nonnegative random variable). For any given integer n 1 and time t > 0, the nth arrival epoch, Sn , and the counting random variable, N (t), are related by {Sn  t} = {N (t) n}. (2.2) To see this, note that {Sn  t} is the event that the nth arrival occurs by time t. This event implies that N (t), the number of arrivals by time t, must be at least n; i.e., it implies the event {N (t) n}. Similarly, {N (t) n} implies {Sn  t}, yielding the equality in (2.2). This equation is essentially obvious from Figure 2.1, but is one of those peculiar obvious things that is often difficult to see. An alternate form, which is occasionally more transparent, comes from taking the complement of both sides of (2.2), getting {Sn > t} = {N (t) < n}. (2.3) For example, the event {S1 > t} means that the first arrival occurs after t, which means {N (t) < 1} (i.e., {N (t) = 0}). These relations will be used constantly in going back and forth between arrival epochs and counting rv’s. In principle, (2.2) or (2.3) can be used to specify joint distribution functions of arrival epochs in terms of joint distribution functions of counting variables and vice versa, so either characterization can be used to specify an arrival process. In summary, then, an arrival process can be specified by the joint distributions of the arrival epochs, the interarrival intervals, or the counting rv’s. In principle, specifying any one of these specifies the others also.3 2.2 Definition and properties of a Poisson process A Poisson process is an example of an arrival process, and the interarrival times provide the most convenient description since the interarrival times are defined to be IID. Processes with IID interarrival times are particularly important and form the topic of Chapter 3. Definition 2.2.1. A renewal process is an arrival process for which the sequence of interarrival times is a sequence of IID rv’s. Definition 2.2.2. A Poisson process is a renewal process in which the interarrival intervals 3 By definition, a stochastic process is a collection of rv’s, so one might ask whether an arrival process (as a stochastic process) is ‘really’ the arrival epoch process 0  S1  S2  · · · or the interarrival process X1 , X2 , . . . or the counting process {N (t); t > 0}. The arrival time process comes to grips with the actual arrivals, the interarrival process is often the simplest, and the counting process ‘looks’ most like a stochastic process in time since N (t) is a rv for each t 0. It seems preferable, since the descriptions are so clearly equivalent, to view arrival processes in terms of whichever description is most convenient. CHAPTER 2. POISSON PROCESSES have an exponential distribution function; i.e., for some real fX (x) = exp( x) for x 0. > 0, each Xi has the density4 The parameter is called the rate of the process. We shall see later that for any interval of size t, t is the expected number of arrivals in that interval. Thus is called the arrival rate of the process. 2.2.1 Memoryless property What makes the Poisson process unique among renewal processes is the memoryless property of the exponential distribution. Definition 2.2.3. Memoryless random variables: A rv X possesses the memoryless property if Pr{X > 0} = 1, (i.e., X is a positive rv) and, for every x 0 and t 0, Pr{X > t + x} = Pr{X > x} Pr{X > t} . (2.4) Note that (2.4) is a statement about the complementary distribution function of X. There is no intimation that the event {X > t + x} in the equation has any particular relation to the events {X > t} or {X > x}. For an exponential rv X of rate > 0, Pr{X > x} = e x for x 0. This satisfies (2.4) for all x 0, t 0, so X is memoryless. Conversely, an arbitrary rv X is memoryless only if it is exponential. To see this, let h(x) = ln[Pr{X > x}] and observe that since Pr{X > x} is nonincreasing in x, h(x) is also. In addition, (2.4) says that h(t + x) = h(x) + h(t) for all x, t 0. These two statements (see Exercise 2.6) imply that h(x) must be linear in x, and Pr{X > x} must be exponential in x. Since a memoryless rv X must be exponential, Pr{X > t} > 0 for all t that we can rewrite (2.4) as Pr{X > t + x | X > t} = Pr{X > x} . 0. This means (2.5) If X is interpreted as the waiting time until some given arrival, then (2.5) states that, given that the arrival has not occurred by time t, the distribution of the remaining waiting time (given by x on the left side of (2.5)) is the same as the original waiting time distribution (given on the right side of (2.5)), i.e., the remaining waiting time has no ‘memory’ of previous waiting. Example 2.2.1. Suppose X is the waiting time, starting at time 0, for a bus to arrive, and suppose X is memoryless. After waiting from 0 to t, the distribution of the remaining waiting time from t is the same as the original distribution starting from 0. The still waiting customer is, in a sense, no better o↵ at time t than at time 0. On the other hand, if the bus is known to arrive regularly every 16 minutes, then it will certainly arrive within a minute, 4 With this density, Pr{Xi >0} = 1, so that we can regard Xi as a positive random variable. Since events of probability zero can be ignored, the density exp( x) for x 0 and zero for x < 0 is e↵ectively the same as the density exp( x) for x > 0 and zero for x  0. 2.2. DEFINITION AND PROPERTIES OF A POISSON PROCESS and X is not memoryless. The opposite situation is also possible. If the bus frequently breaks down, then a 15 minute wait can indicate that the remaining wait is probably very long, so again X is not memoryless. We study these non-memoryless situations when we study renewal processes in the next chapter. Although memoryless distributions must be exponential, it can be seen that if the definition of memoryless is restricted to integer times, then the geometric distribution becomes memoryless, and it can be seen as before that this is the only memoryless integer-time distribution. In this respect, the Bernoulli process (which has geometric interarrival times) is like a discretetime version of the Poisson process (which has exponential interarrival times). We now use the memoryless property of exponential rv’s to find the distribution of the first arrival in a Poisson process after an arbitrary given time t > 0. We not only find this distribution, but also show that this first arrival after t is independent of all arrivals up to and including t. More precisely, we prove the following theorem. Theorem 2.2.1. For a Poisson process of rate , and any given t > 0, the length of the interval from t until the first arrival after t is a nonnegative rv Z with the distribution function 1 exp[ z] for z 0. This rv is independent of all arrival epochs before time t and independent of the set of rv’s {N(⌧); ⌧  t}. The basic idea behind this theorem is to note that Z, conditional on the time ⌧ of the last arrival before t, is simply the remaining time until the next arrival. Since the interarrival time starting at ⌧ is exponential and thus memoryless, Z is independent of ⌧  t, and of all earlier arrivals. The following proof carries this idea out in detail. Z X1 0 t - S1 X2 - S2 Figure 2.2: For arbitrary fixed t > 0, consider the event N (t) = 0. Conditional on this event, Z is the distance from t to S1 ; i.e., Z = X1 t. Proof: Let Z be the distance from t until the first arrival after t. We first condition on N (t) = 0 (see Figure 2.2). Given N (t) = 0, we see that X1 > t and Z = X1 t. Thus, Pr{Z > z | N (t)=0} = Pr{X1 > z + t | N (t)=0} = Pr{X1 > z + t | X1 > t} = Pr{X1 > z} = e z . (2.6) (2.7) In (2.6), we used the fact that {N (t) = 0} = {X1 > t}, which is clear from Figure 2.1 (and also from (2.3)). In (2.7) we used the memoryless condition in (2.5) and the fact that X1 is exponential. CHAPTER 2. POISSON PROCESSES Next consider the conditions that N (t) = n (for arbitrary n > 1) and Sn = ⌧ (for arbitrary ⌧  t). The argument here is basically the same as that with N (t) = 0, with a few extra details (see Figure 2.3). X3 - N (t) X2 - 6 Z - X1 0 S1 ⌧ S2 t S3 Figure 2.3: Given N (t) = 2, and S2 = ⌧ , X3 is equal to Z + (t ⌧ ). Also, the event {N (t)=2, S2 =⌧ } is the same as the event {S2 =⌧, X3 >t ⌧ }. Conditional on N (t) = n and Sn = ⌧ , the first arrival after t is the first arrival after the arrival at Sn , i.e., Z = z corresponds to Xn+1 = z + (t ⌧ ). Pr{Z > z | N (t)=n, Sn =⌧ } = Pr{Xn+1 > z+t ⌧ | N (t)=n, Sn =⌧ } = Pr{Xn+1 > z+t ⌧ | Xn+1 >t ⌧, Sn =⌧ } = Pr{Xn+1 > z+t ⌧ | Xn+1 >t ⌧ } = Pr{Xn+1 > z} = e z , (2.8) (2.9) (2.10) (2.11) where (2.9) follows because, given Sn = ⌧  t, we have {N (t) = n} = {Xn+1 > t ⌧ } (see Figure 2.3). Eq. (2.10) follows because Xn+1 is independent of Sn . Eq. (2.11) follows from the memoryless condition in (2.5) and the fact that Xn+1 is exponential. The same argument applies if, in (2.8), we condition not only on Sn but also on S1 , . . . , Sn Since this is equivalent to conditioning on N (⌧ ) for all ⌧ in (0, t], we have Pr{Z > z | {N (⌧ ), 0 < ⌧  t}} = exp( z). 1. (2.12) Next consider subsequent interarrival intervals after a given time t. For m 2, let Zm be the interarrival interval from the m 1st arrival epoch after t to the mth arrival epoch after t. Let Z in (2.12) be denoted as Z1 here. Given N (t) = n and Sn = ⌧ , we see that Zm = Xm+n for m 2, and therefore Z1 , Z2 , . . . , are IID exponentially distributed rv’s, conditional on N (t) = n and Sn = ⌧ (see Exercise 2.8). Since this is independent of N (t) and Sn , we see that Z1 , Z2 , . . . are unconditionally IID and also independent of N (t) and Sn . It should also be clear that Z1 , Z2 , . . . are independent of {N (⌧ ); 0 < ⌧  t}. The above argument shows that the portion of a Poisson process starting at an arbitrary time t > 0 is a probabilistic replica of the process starting at 0; that is, the time until the first arrival after t is an exponentially distributed rv with parameter , and all subsequent arrivals are independent of this first arrival and of each other and all have the same exponential distribution. 2.2. DEFINITION AND PROPERTIES OF A POISSON PROCESS Definition 2.2.4. A counting process {N (t); t > 0} has the stationary increment property if for every t0 > t > 0, N (t0 ) N (t) has the same distribution function as N (t0 t). 0 ] for any e 0 0 0 e Let ustdefine given t. N (t, t ) = N (t ) N (t) as the number of arrivals in the interval (t, t We have just shown that for a Poisson process, the rv N(t, t0) has the same distribution as N(t0 t), which means that a Poisson process has the stationary increment property. Thus,but thenot distribution of the point. number of arrivals in an interval depends on the size of the interval on its starting Definition 2.2.5. A counting process {N (t); t > 0} has the independent increment prop­ erty if, for every integer k > 0, and every k-tuple of times 0 < t1 < t2 < · · · < tk , the e (t1 , t2 ), . . . , N e (tk 1 , tk ) of rv’s are statistically independent. k-tuple of rv’s N (t1 ), N For the Poisson process, Theorem 2.2.1 says that for any t, the time Z1 until the next arrival after t is independent of N (⌧ ) for all ⌧  t. Letting t1 < t2 < · · · tk 1 < e (t1 , t2 ), . . . , N e (tk 1 , t). We have also t, this means that Z1 is independent of N (t1 ), N e (t, t0 ) are independent seen that the subsequent interarrival times after Z1 , and thus N 0 e (tk , tk+1 ) e e of N (t1 ), N (t1 , t2 ), . . . , N (tk 1 , t). Renaming t as tk and t as tk+1 , we see that N e (t1 , t2 ), . . . , N e (tk 1 , tk ). Since this is true for all k, the Poisson is independent of N (t1 ), N process has the independent increment property. In summary, we have proved the following: Theorem 2.2.2. Poisson processes have both the stationary increment and independent increment properties. Note that if we look only at integer times, then the Bernoulli process also has the stationary and independent increment properties. 2.2.2 Probability density of Sn and S1 , . . . Sn Recall from (2.1) that, for a Poisson process, Sn is the sum of n IID rv’s, each with the density function fX (x) = exp( x), x 0. Also recall that the density of the sum of two independent rv’s can be found by convolving their densities, and thus the density of S2 can be found by convolving fX (x) with itself, S3 by convolving the density of S2 with fX (x), and so forth. The result, for t 0, is called the Erlang density,5 fSn (t) = n tn 1 exp( (n t) 1)! (2.13) . We can understand this density (and other related matters) much better by reviewing the above mechanical derivation more carefully. The joint density for two continuous indepen­ dent rv’s X1 and X2 is given by fX1 X2 (x1 , x2 ) = fX1 (x1 )fX2 (x2 ). Letting S2 = X1 + X2 and substituting S2 X1 for X2 , we get the following joint density for X1 and the sum S2 , fX1 S2 (x1 , s2 ) = fX1 (x1 )fX2 (s2 5 x1 ). Another (somewhat rarely used) name for the Erlang density is the gamma density. CHAPTER 2. POISSON PROCESSES The marginal density for S2 then results from integrating x1 out from the joint density, and this, of course, is the familiar convolution integration. For IID exponential rv’s X1 , X2 , the joint density of X1 , S2 takes the following interesting form: fX1 S2 (x1 s2 ) = 2 exp( x1 ) exp( (s2 x1 )) = 2 exp( s2 ) for 0  x1  s2 . (2.14) This says that the joint density does not contain x1 , except for the constraint 0  x1  s2 . Thus, for fixed s2 , the joint density, and thus the conditional density of X1 given S2 = s2 is uniform over 0  x1  s2 . The integration over x1 in the convolution equation is then simply multiplication by the interval size s2 , yielding the marginal distribution fS2 (s2 ) = 2 s exp( s2 ), in agreement with (2.13) for n = 2. 2 This same curious behavior exhibits itself for the sum of an arbitrary number n of IID exponential rv’s. That is, fX1 ···Xn (x1 , . . . , xn ) = n exp( x1 x2 · · · xn ). Letting Sn = X1 + · · · + Xn and substituting Sn X1 · · · Xn 1 for Xn , this becomes fX1 ···Xn 1 Sn (x1 , . . . , xn 1 , sn ) = n exp( sn ). since each xi cancels out above. This equation is valid over the region where each xi and sn x1 · · · xn 1 0. The density is 0 elsewhere. 0 The constraint region becomes more clear here if we replace the interarrival intervals X1 , . . . , Xn 1 with the arrival epochs S1 , . . . , Sn 1 where S1 = X1 and Si = Xi + Si 1 for 2  i  n 1. The joint density then becomes6 fS1 ···Sn (s1 , . . . , sn ) = n exp( sn ) for 0  s1  s2 · · ·  sn . (2.15) The interpretation here is the same as with S2 . The joint density does not contain any arrival time other than sn , except for the ordering constraint 0  s1  s2  · · ·  sn , and thus this joint density is constant over all choices of arrival times satisfying the ordering constraint. Mechanically integrating this over s1 , then s2 , etc. we get the Erlang formula (2.13). The Erlang density then is the joint density in (2.15) times the volume snn 1 /(n 1)! of the region of s1 , . . . , sn 1 satisfing 0 < s1 < · · · < sn . This will be discussed further later. Note that (2.15), for all n specifies the joint distribution for all arrival times, and thus fully specifes a Poisson process. An alternate definition for the Poisson process is then any process whose joint arrival time distribution satisifes (2.15). This is not customarily used to define the Poisson process, whereas two alternate definitions given subsequently often are used as a starting definition. 2.2.3 The PMF for N (t) The Poisson counting process, {N (t); t > 0} consists of a nonnegative integer rv N (t) for each t > 0. In this section, we show that the PMF for this rv is the well-known Poisson 6 The random vector S = (S1 , . . . , Sn ) is then related to the interarrival intervals X = (X1 , . . . , Xn ) by a linear transformation, say S = AX where A is an upper triangular matrix with ones on the main diagonal and on all elements above the main diagonal. In general, the joint density of a non-singular linear transformation AX at X = x is fX (x )/| det A|. This is because the transformation A carries an incremental cube, on each side, into a parallelepiped of volume n | det A|. Since, for the case here, A is upper triangular with 1’s on the diagonal, det A = 1. 2.2. DEFINITION AND PROPERTIES OF A POISSON PROCESS PMF, as stated in the following theorem. We give two proofs for the theorem, each providing its own type of understanding and each showing the close relationship between {N(t) = n} and {Sn = t}. Theorem 2.2.3. For a Poisson process of rate , and for any t > 0, the PMF for N (t) (i.e., the number of arrivals in (0, t]) is given by the Poisson PMF, pN (t) (n) = ( t)n exp( n! t) (2.16) . Proof 1: This proof, for given n and t, is based on two ways of calculating Pr{t < Sn+1  t + } for some vanishingly small . The first way is based on the already known density of Sn+1 and gives Z t+ Pr{t < Sn+1  t + } = fSn (⌧ ) d⌧ = fSn (t) ( + o( )). t The term o( ) is used to describe a function of that goes to 0 faster than as ! 0. More precisely, a function g( ) is said to be of order o( ) if lim !0 g( ) = 0. Thus Pr{t < Sn  t + } = fSn (t)( + o( )) is simply a consequence of the fact that Sn has a continuous probability density in the interval [t, t + ]. The second way to calculate Pr{t < Sn+1  t + } is to first observe that the probability of more than 1 arrival in (t, t + ]) is o( ). Ignoring this possibility, {t < Sn+1  t + } occurs if exactly n arrivals are in the interval (0, t] and one arrival occurs in (t, t + ]. Because of the independent increment property, this is an event of probability pN (t) (n)( + o( )). Thus pN (t) (n)( Dividing by + o( )) + o( ) = fSn+1 (t)( + o( )). and taking the limit ! 0, we get pN (t) (n) = fSn+1 (t). Using the density for fSn in (2.13), we get (2.16). Proof 2: The approach here is to use the fundamental relation that {N (t) Taking the probabilities of these events, 1 X i=n pN (t) (i) = Z 0 t fSn (⌧ ) d⌧ for all n n} = {Sn  t}. 1 and t > 0. The term on the right above is the distribution function of Sn and the term on the left is the complementary distribution function of N (t). The complementary distribution function and the PMF of N (t) uniquely specify each other, so the theorem is equivalent to showing that X1 ( t)i exp( t) Z t = fSn (⌧ ) d⌧. (2.17) i=n i! 0 CHAPTER 2. POISSON PROCESSES If we take the derivative with respect to t of each side of (2.17), we find that almost magically each term except the first on the left cancels out, leaving us with n tn 1 exp( (n t) 1)! = fSn (t). Thus the derivative with respect to t of each side of (2.17) is equal to the derivative of the other for all n 1 and t > 0. The two sides of (2.17) are also equal in the limit t ! 0, so it follows that (2.17) is satisfied everywhere, completing the proof. 2.2.4 Alternate definitions of Poisson processes Definition 2 of a Poisson process: A Poisson counting process {N (t); t > 0} is a counting process that satisfies (2.16) (i.e., has the Poisson PMF) and has the independent and stationary increment properties. We have seen that the properties in Definition 2 are satisfied starting with Definition 1 (using IID exponential interarrival times), so Definition 1 implies Definition 2. Exercise 2.4 shows that IID exponential interarrival times are implied by Definition 2, so the two definitions are equivalent. It may be somewhat surprising at first to realize that a counting process that has the Poisson PMF at each t is not necessarily a Poisson process, and that the independent and stationary increment properties are also necessary. One way to see this is to recall that the Poisson PMF for all t in a counting process is equivalent to the Erlang density for the successive arrival epochs. Specifying the probability density for S1 , S2 , . . . , as Erlang specifies the marginal densities of S1 , S2 , . . . , but need not specify the joint densities of these rv’s. Figure 2.4 illustrates this in terms of the joint density of S1 , S2 , given as fS1 S2 (s1 s2 ) = 2 exp( s2 ) for 0  s1  s2 and 0 elsewhere. The figure illustrates how the joint density can be changed without changing the marginals. There is a similar e↵ect with the Bernoulli process in that a discrete counting process for which the number of arrivals from 0 to t, for each integer t, is a binomial rv, but the process is not Bernoulli. This is explored in Exercise 2.5. The next definition of a Poisson process is based on its incremental properties. Consider e (t, t + ) has the same the number of arrivals in some very small interval (t, t + ]. Since N distribution as N ( ), we can use (2.16) to get n o e (t, t + ) = 0 Pr N = e ⇡ 1 + o( ) n o e (t, t + ) = 1 Pr N = e ⇡ + o( ) o n e (t, t + ) 2 Pr N ⇡ o( ). (2.18) Definition 3 of a Poisson process: A Poisson counting process is a counting process that satisfies (2.18) and has the stationary and independent increment properties. 2.2. DEFINITION AND PROPERTIES OF A POISSON PROCESS ⌘ ⌘ ⌘ ⌘ s1 ⌘ ⌘ ⌘ ⌘ ⌘ ⌘ 0 fS1 S2 (s1 s2 ) > 0 s2 Figure 2.4: The joint density of S1 , S2 is nonzero in the region shown. It can be changed, while holding the marginals constant, by reducing the joint density by ✏ in the upper left and lower right squares above and increasing it by ✏ in the upper right and lower left squares. We have seen that Definition 1 implies Definition 3. The essence of the argument the other way is that for any interarrival interval X, FX (x + ) FX (x) is the probability of an arrival in an appropriate infinitesimal interval of width , which by (2.18) is +o( ). Turning this into a di↵erential equation (see Exercise 2.7), we get the desired exponential interarrival intervals. Definition 3 has an intuitive appeal, since it is based on the idea of independent arrivals during arbitrary disjoint intervals. It has the disadvantage that one must do a considerable amount of work to be sure that these conditions are mutually consistent, and probably the easiest way is to start with Definition 1 and derive these properties. Showing that there is a unique process that satisfies the conditions of Definition 3 is even harder, but is not necessary at this point, since all we need is the use of these properties. Section 2.2.5 will illustrate better how to use this definition (or more precisely, how to use (2.18)). What (2.18) accomplishes in Definition 3, beyond the assumption of independent and sta­ tionary increments, is the prevention of bulk arrivals. For example, consider a counting process in which arrivals always occur in pairs, and the intervals between successive pairs arenIID and exponentially distributed with parameter (see Figure 2.5). For this process, o n o e e Pr N (t, t + )=1 = 0, and Pr N (t, t+ )=2 = + o( ), thus violating (2.18). This process has stationary and independent increments, however, since the process formed by viewing a pair of arrivals as a single incident is a Poisson process. 2.2.5 The Poisson process as a limit of shrinking Bernoulli processes The intuition of Definition 3 can be achieved in a less abstract way by starting with the Bernoulli process, which has the properties of Definition 3 in a discrete-time sense. We then go to an appropriate limit of a sequence of these processes, and find that this sequence of Bernoulli processes converges in some sense to the Poisson process. Recall that a Bernoulli process is an IID sequence, Y1 , Y2 , . . . , of binary random variables for which pY (1) = p and pY (0) = 1 p. We can visualize Yi = 1 as an arrival at time i and Yi = 0 as no arrival, but we can also ‘shrink’ the time scale of the process so that for some integer j > 0, Yi is an arrival or no arrival at time i2 j . We consider a sequence indexed CHAPTER 2. POISSON PROCESSES N (t) 4 X2 2 X1 - 3 - 1 0 S1 S2 Figure 2.5: A counting process modeling bulk arrivals. X1 is the time until the first pair of arrivals and X2 is the interval between the first and second pair of arrivals. by j of such shrinking Bernoulli processes, and in order to keep the arrival rate constant, we let p = 2 j for the jth process. Thus for each unit increase in j, the Bernoulli process shrinks by replacing each slot with two slots, each with half the previous arrival probability. The expected number of arrivals per unit time is then , matching the Poisson process that we are approximating. If we look at this jth process relative to Definition 3 of a Poisson process, we see that for these regularly spaced increments of size = 2 j , the probability of one arrival in an increment is and that of no arrival is 1 , and thus (2.18) is satisfied, and in fact the o( ) terms are exactly zero. For arbitrary sized increments, it is clear that disjoint increments have independent arrivals. The increments are not quite stationary, since, for example, an increment of size 2 j 1 might contain a time that is a multiple of 2 j or might not, depending on its placement. However, for any fixed increment of size , the number of multiples of 2 j (i.e., the number of possible arrival points) is either b 2j c or 1 + b 2j c. Thus in the limit j ! 1, the increments are both stationary and independent. For each j, the jth Bernoulli process has an associated Bernoulli counting process Nj (t) = Pbt2j c i=1 Yi . This is the number of arrivals up to time t and is a discrete rv with the binomial j j PMF. That is, pNj (t) (n) = btn2 c pn (1 p)bt2 c n where p = 2 j . We now show that this PMF approaches the Poisson PMF as j increases.7 Theorem 2.2.4. Consider the sequence of shrinking Bernoulli processes with arrival prob­ ability 2 j and time-slot size 2 j . Then for every fixed time t > 0 and fixed number of arrivals n, the counting PMF pNj (t) (n) approaches the Poisson PMF (of the same ) with increasing j, i.e., lim pNj (t) (n) = pN (t) (n). j!1 7 (2.19) This limiting result for the binomial distribution is very di↵erent from the asymptotic results in Chapter 1 for the binomial. Here the parameter p of the binomial is shrinking with increasing j, whereas there, p is constant while the number of variables is increasing. 2.2. DEFINITION AND PROPERTIES OF A POISSON PROCESS Proof: We first rewrite the binomial PMF, for bt2j c variables with p = 2 j as ✓ j ◆✓ ◆n bt2 c 2 j lim pNj (t) (n) = lim exp[bt2j c(ln(1 2 j )] j!1 j!1 1 n 2 j ✓ j ◆✓ ◆n bt2 c 2 j = lim (2.20) exp( t) j!1 n 1 2 j ◆n ✓ bt2j c · bt2j 1c · · · bt2j n+1c 2 j = lim exp( t) (2.21) j!1 n! 1 2 j ( t)n exp( t) . = (2.22) n! We used ln(1 2 j) = 2 j + o(2 j ) in (2.20) and expanded ⇣ ⌘ the combinatorial term 2 j j ic 1 2 j = t for 0  i  n 1. in (2.21). In (2.22), we recognized that limj!1 bt2 Since the binomial PMF (scaled as above) has the Poisson PMF as a limit for each n, the distribution function of Nj (t) also converges to the Poisson distribution function for each t. In other words, for each t > 0, the counting random variables Nj (t) of the Bernoulli processes converge in distribution to N (t) of the Poisson process. This does not say that the Bernoulli counting processes converge to the Poisson counting process in any meaningful sense, since the joint distributions are also of concern. The following corollary treats this. Corollary 2.2.1. For any finite integer k > 0, let 0 < t1 < t2 < · · · < tk be any set of time instants. Then the joint distribution function of Nj (t1 ), Nj (t2 ), . . . Nj (tk ) approaches the joint distribution function of N (t1 ), N (t2 ), . . . N (tk ) as j ! 1. Proof: It is sufficient to show that the joint PMF’s converge. We can rewrite the joint PMF for each Bernoulli process as pNj (t1 ),... ,Nj (tk ) (n1 , . . . , nk ) = pNj (t1 ),Ñj (t1 ,t2 ),... ,Ñj (tk = pNj (t1 ) (n1 ) k Y pÑj (t` ,t` 1 ,tk ) 1) (n1 , n2 n1 , . . . , nk nk (n` n` 1) 1) (2.23) `=2 where we have used the independent increment property for the Bernoulli process. For the Poisson process, we similarly have pN (t1 ),... ,N (tk ) (n1 , . . . nk ) = pN (t1 ) (n1 ) k Y pÑ (t` ,t` 1) (n` n` 1) (2.24) `=2 Taking the limit of (2.23) as j ! 1, we recognize from Theorem 2.2.4 that each term of (2.23) goes to the corresponding term in (2.24). For the Ñ rv’s , this requires a trivial generalization in Theorem 2.2.4 to deal with the arbitrary starting time. We conclude from this that the sequence of Bernoulli processes above converges to the Poisson process in the sense of the corollary. Recall from Section 1.5.5 that there are a CHAPTER 2. POISSON PROCESSES number of ways in which a sequence of rv’s can converge. As one might imagine, there are many more ways in which a sequence of stochastic processes can converge, and the corollary simply establishes one of these. We have neither the mathematical tools nor the need to delve more deeply into these convergence issues. Both the Poisson process and the Bernoulli process are so easy to analyze that the con­ vergence of shrinking Bernoulli processes to Poisson is rarely the easiest way to establish properties about either. On the other hand, this convergence is a powerful aid to the in­ tuition in understanding each process. In other words, the relation between Bernoulli and Poisson is very useful in suggesting new ways of looking at problems, but is usually not the best way to analyze those problems. 2.3 Combining and splitting Poisson processes Suppose that ; : t > 0} and {N2 (t); t > 0} are independent Poisson counting processes8 of rates 1 and 2 respectively. We want to look at the sum process where N (t) = N1 (t)+N2 (t) for all t 0. In other words, {N (t); t > 0} is the process consisting of all arrivals to both process 1 and process 2. We shall show that {N (t); t > 0} is a Poisson counting process of rate = 1 + 2 . We show this in three di↵erent ways, first using Definition 3 of a Poisson process (since that is most natural for this problem), then using Definition 2, and finally Definition 1. We then draw some conclusions about the way in which each approach is helpful. Since {N1 (t); t > 0} and {N2 (t); t > 0} are independent and each possess the stationary and independent increment properties, it follows from the definitions that {N (t); t > 0} also possesses the stationary and independent increment properties. Using the approximations in (2.18) for the individual processes, we see that n o n o n o e (t, t + ) = 0 e1 (t, t + ) = 0 Pr N e2 (t, t + ) = 0 Pr N = Pr N = (1 )⇡1 . n o e (t, t+ ) = 1 is approximated by where 1 2 2 has been dropped. In the same way, Pr N n o e (t, t + ) 2 is approximated by 0, both with errors proportional to 2 . It and Pr N follows that {N (t); t > 0} is a Poisson process. 1 )(1 2 In the second approach, we have N (t) = N1 (t) + N2 (t). Since N (t), for any given t, is the sum of two independent Poisson rv’s , it is also a Poisson rv with mean t = 1 t + 2 t. If the reader is not aware that the sum of two independent Poisson rv’s is Poisson, it can be derived by discrete convolution of the two PMF’s (see Exercise 1.19). More elegantly, one can observe that we have already implicitly shown this fact. That is, if we break an interval I into disjoint subintervals, I1 and I2 , then the number of arrivals in I (which is Poisson) is the sum of the number of arrivals in I1 and in I2 (which are independent Poisson). 8 Two processes {N1 (t); t > 0} and {N2 (t); t > 0} are said to be independent if for all positive integers k and all sets of times 0 < t1 < t2 < · · · < tk , the random variables N1 (t1 ), . . . , N1 (tk ) are independent of N2 (t1 ), . . . , N2 (tk ). Here it is enough to extend the independent increment property to independence between increments over the two processes; equivalently, one can require the interarrival intervals for one process to be independent of the interarrivals for the other process. 2.3. COMBINING AND SPLITTING POISSON PROCESSES Finally, since N (t) is Poisson for each t, and since the stationary and independent increment properties are satisfied, {N (t); t > 0} is a Poisson process. In the third approach, X1 , the first interarrival interval for the sum process, is the minimum of X11 , the first interarrival interval for the first process, and X21 , the first interarrival interval for the second process. Thus X1 > t if and only if both X11 and X21 exceed t, so Pr{X1 > t} = Pr{X11 > t} Pr{X21 > t} = exp( 1t 2 t) = exp( t). Using the memoryless property, each subsequent interarrival interval can be analyzed in the same way. The first approach above was the most intuitive for this problem, but it required constant care about the order of magnitude of the terms being neglected. The second approach was the simplest analytically (after recognizing that sums of independent Poisson rv’s are Pois­ son), and required no approximations. The third approach was very simple in retrospect, but not very natural for this problem. If we add many independent Poisson processes to­ gether, it is clear, by adding them one at a time, that the sum process is again Poisson. What is more interesting is that when many independent counting processes (not necessar­ ily Poisson) are added together, the sum process often tends to be approximately Poisson if the individual processes have small rates compared to the sum. To obtain some crude intu­ ition about why this might be expected, note that the interarrival intervals for each process (assuming no bulk arrivals) will tend to be large relative to the mean interarrival interval for the sum process. Thus arrivals that are close together in time will typically come from di↵erent processes. The number of arrivals in an interval large relative to the combined mean interarrival interval, but small relative to the individual interarrival intervals, will be the sum of the number of arrivals from the di↵erent processes; each of these is 0 with large probability and 1 with small probability, so the sum will be approximately Poisson. 2.3.1 Subdividing a Poisson process Next we look at how to break {N (t); t > 0}, a Poisson counting process of rate , into two processes, {N1 (t); t > 0} and {N2 (t); t > 0}. Suppose that each arrival in {N (t); t > 0} is sent to the first process with probability p and to the second process with probability 1 p (see Figure 2.6). Each arrival is switched independently of each other arrival and independently of the arrival epochs. It may be helpful to visualize this as the combination of two independent processes. The first is the Poisson process of rate and the second is a Bernoulli process Xn ; n 1} where pXn (1) = p and pXn (2) = 1 p. The nth arrival of the Poisson process is then labeled as a type 1 arrival if Xn = 1 and as a type 2 arrival with probability 1 p. We shall show that the resulting processes are each Poisson, with rates 1 = p and 2 = (1 p) respectively, and that furthermore the two processes are independent. Note that, conditional on the original process, the two new processes are not independent; in fact one completely determines the other. Thus this independence might be a little surprising. First consider a small increment (t, t + ]. The original process has an arrival in this incremental interval with probability (ignoring 2 terms as usual), and thus process 1 CHAPTER 2. POISSON PROCESSES N (t) rate p -H HH1 p HH N1 (t) rate 1 = p - N2 (t) rate 2 = (1 p) Figure 2.6: Each arrival is independently sent to process 1 with probability p and to process 2 otherwise. has an arrival with probability p and process 2 with probability (1 p). Because of the independent increment property of the original process and the independence of the division of each arrival between the two processes, the new processes each have the independent increment property, and from above have the stationary increment property. Thus each process is Poisson. Note now that we cannot verify that the two processes are independent from this small increment model. We would have to show that the number of arrivals for process 1 and 2 are independent over (t, t + ]. Unfortunately, leaving out the terms of order 2 , there is at most one arrival to the original process and no possibility of an arrival to each new process in (t, t + ]. If it is impossible for both processes to have an arrival in the same interval, they cannot be independent. It is possible, of course, for each process to have an arrival in the same interval, but this is a term of order 2 . Thus, without paying attention to the terms of order 2 , it is impossible to demonstrate that the processes are independent. To demonstrate that process 1 and 2 are independent, we first calculate the joint PMF for N1 (t), N2 (t) for arbitrary t. Conditioning on a given number of arrivals N (t) for the original process, we have Pr{N1 (t)=m, N2 (t)=k | N (t)=m+k} = (m + k)! m p (1 m!k! p)k . (2.25) Equation (2.25) is simply the binomial distribution, since, given m + k arrivals to the original process, each independently goes to process 1 with probability p. Since the event {N1 (t) = m, N2 (t) = k} is a subset of the conditioning event above, Pr{N1 (t)=m, N2 (t)=k | N (t)=m+k} = Pr{N1 (t)=m, N2 (t)=k} . Pr{N (t)=m+k} Combining this with (2.25), we have Pr{N1 (t)=m, N2 (t)=k} = (m + k!) m p (1 m!k! p)k ( t)m+k e t . (m + k)! (2.26) Rearranging terms, we get Pr{N1 (t)=m, N2 (t)=k} = (p t)m e m! pt [(1 p) t]k e k! (1 p)t . (2.27) 2.3. COMBINING AND SPLITTING POISSON PROCESSES This shows that N1 (t) and N2 (t) are independent. To show that the processes are indepen­ dent, we must show that for any k > 1 and any set of times 0  t1  t2  · · ·  tk , the sets {N1 (ti ); 1  i  k} and {N2 (tj ); 1  j  k} are independent of each other. It is equivalent e1 (ti 1 , ti ); 1  i  k} and {N e2 (tj 1 , tj ); 1  j  k} (where t0 is 0) to show that the sets {N are independent. The argument above shows this independence for i = j, and for i = 6 j, the independence follows from the independent increment property of {N (t); t > 0}. 2.3.2 Examples using independent Poisson processes We have observed that if the arrivals of a Poisson process are split into two new arrival processes, each arrival of the original process independently going into the first of the new processes with some fixed probability p, then the new processes are Poisson processes and are independent. The most useful consequence of this is that any two independent Poisson processes can be viewed as being generated from a single process in this way. Thus, if one process has rate 1 and the other has rate 2 , they can be viewed as coming from a process of rate 1 + 2 . Each arrival to the combined process is then labeled as a first process arrival with probability p = 1 /( 1 + 2 ) and as a second process arrival with probability 1 p. The above point of view is very useful for finding probabilities such as Pr{S1k < S2j } where S1k is the epoch of the kth arrival to the first process and S2j is the epoch of the jth arrival to the second process. The problem can be rephrased in terms of a combined process to ask: out of the first k + j 1 arrivals to the combined process, what is the probability that k or more of them are switched to the first process? (Note that if k or more of the first k + j 1 go to the first process, at most j 1 go to the second, so the kth arrival to the first precedes the jth arrival to the second; similarly if fewer than k of the first k + j 1 go to the first process, then the jth arrival to the second process precedes the kth arrival to the first). Since each of these first k + j 1 arrivals are switched independently with the same probability p, the answer is Pr{S1k < S2j } = Xk+j i=k 1 (k + j 1)! pi (1 i!(k + j 1 i)! p)k+j 1 i . (2.28) Example 2.3.1. [The M/M/1 queue] Queueing theorists use a standard notation of char­ acters separated by slashes to describe common types of queueing systems. The first char­ acter 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 an 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 which server is used. Consider an M/M/1 queue, i.e., a queueing system with a Poisson arrival system (say of rate ) and a single server who serves arriving customers in order with a service time distribution F(y) = 1 exp[ µy]. Thus during periods when the server is busy, customers CHAPTER 2. POISSON PROCESSES leave the system according to a Poisson process (process 2) of rate µ. We then see that if j or more customers are waiting at a given time, then (2.28) gives the probability that the kth subsequent arrival comes before the jth departure. 2.4 Non-homogeneous Poisson processes The Poisson process, as we defined it, is characterized by a constant arrival rate . It is often useful to consider a more general type of process in which the arrival rate varies as a function of time. A non-homogeneous Poisson process with time varying arrival rate (t) is defined9 as a counting process {N (t); t > 0} which has the independent increment property and, for all t 0, > 0, also satisfies: n o e (t, t + ) = 0 Pr N = 1 (t) + o( ) n o e (t, t + ) = 1 Pr N = (t) + o( ) n o e (t, t + ) 2 Pr N = o( ). (2.29) e (t, t + ) = N (t + ) N (t). The non-homogeneous Poisson process does not have where N the stationary increment property. One common application occurs in optical communication where a non-homogeneous Pois­ son 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 Pois­ son process. To see how to do this, assume that (t) is bounded away from zero. We partition the time axis into increments whose lengths vary inversely with (t), thus hold­ ing the probability of an arrival in an increment at some fixed value p = (t). Thus, temporarily ignoring the variation of (t) within an increment, ⇢ ✓ ◆ p e Pr N t, t + =0 = 1 p + o(p) (t) ⇢ ✓ ◆ e t, t + p Pr N =1 = p + o(p) (t) ⇢ ✓ ◆ p e Pr N t, t + 2 = o(✏). (2.30) (t) This partition is defined more precisely by defining m(t) as Z t m(t) = (⌧ )d⌧. (2.31) 0 9 We assume that (t) is right continuous, i.e., that for each t, (t) is the limit of (t + ✏) as ✏ 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 (2.29) to talk about the distribution of arrivals just to the right of time t. 2.4. NON-HOMOGENEOUS POISSON PROCESSES Then the ith increment ends at that t for which m(t) = i p. r @ @ (t) @ @ t Figure 2.7: Partitioning the time axis into increments each with an expected number of arrivals equal to p. Each rectangle or trapezoid above has the same area, which ensures that the ith partition ends where m(t) = i p. As before, let {Yi ; i 1} be a sequence of IID binary rv’s with Pr{Yi = 1} = p and Pr{Yi = 0} = 1 p. 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 ) = i p. Thus, N (ti ) = Y1 + Y2 + · · · + Yi . If p is decreased as 2 j , each increment is successively split into a pair of increments. Thus by the same argument as in (2.22), [1 + o(p)][m(t)]n exp[ m(t)] . (2.32) n! R⌧ Similarly, for any interval (t, ⌧ ], taking m(t, e ⌧ ) = t (u)du, and taking t = tk , ⌧ = ti for some k, i, we get Pr{N (t) = n} = n o e ⌧ )]n exp[ m(t, e ⌧ )] e (t, ⌧ ) = n = [1 + o(p)][m(t, Pr N . n! (2.33) Going to the limit p ! 0, the counting process {N (t); t > 0} above approaches the non­ homogeneous Poisson process under consideration, and we have the following theorem: Theorem 2.4.1. For a non-homogeneous Poisson process with right-continuous arrival e (t, ⌧ ), the number of arrivals in rate (t) bounded away from zero, the distribution of N (t, ⌧ ], satisfies Z ⌧ n o [m(t, e ⌧ )]n exp[ m e (t, ⌧ )] e Pr N (t, ⌧ ) = n = where m(t, e ⌧) = (u) du. (2.34) n! t 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). Using the queueing notation explained in Ex­ ample 2.3.1, an M/G/1 queue indicates a queue with Poisson arrivals, a general service distribution, and an infinite number of servers. 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 given distribution function G(y); the service time is the interval from start to completion CHAPTER 2. POISSON PROCESSES N (t) - 1 - N1 (⌧ ) = Customers in service at ⌧ G(⌧ t) HH HG(⌧ HH t) - N (⌧ ) N1 (⌧ ) = Customers departed by ⌧ Figure 2.8: Poisson arrivals {N (t); t > 0} can be considered to be split in a non­ homogeneous way. An arrival at t is split with probability 1 of customers still in service at ⌧ . G(⌧ t) into a process of service and is also independent of arrival epochs. We would like to find the distribution function of the number of customers being served at a given epoch ⌧ . Let {N (t); t > 0} be the Poisson counting process, at rate , of customer arrivals. Consider the arrival times of those customers that are still in service at some fixed time ⌧ . In some arbitrarily small interval (t, t+ ], the probability of an arrival is +o( ) and the probability of 2 or more arrivals is negligible (i.e., o( )). The probability that a customer arrives in (t, t + ] and is still being served at time ⌧ > t is then [1 G(⌧ t)] + o( ). Consider a counting process {N1 (t); 0 0} have the independent increment property; also the arrivals in {N (t); t > 0} have independent service times, and thus are independently in or not in {N1 (t); 0 < t  ⌧ }. It follows that {N1 (t); 0 < t  ⌧ } is a non-homogeneous Poisson process with rate [1 G(⌧ t)] at time t  ⌧ . The expected number of arrivals still in service at time ⌧ is then m(⌧ ) = Z ⌧ [1 G(⌧ t=0 t)] dt = Z ⌧ [1 G(t)] dt. (2.35) t=0 and the PMF of the number in service at time ⌧ is given by Pr{N1 (⌧ ) = n} = m(⌧ )n exp( m(⌧ )) . n! (2.36) Note that as ⌧ ! 1, the integral in (2.35) approaches the mean of the service time distri­ bution (i.e., it is the integral of the complementary distribution function, 1 G(t), of the service time). This means that in steady state (as ⌧ ! 1), the distribution of the num­ ber in service at ⌧ depends on the service time distribution only through its mean. This example can be used to model situations such as the number of phone calls taking place at a given epoch. This requires arrivals of new calls to be modeled as a Poisson process and the holding time of each call to be modeled as a random variable independent of other holding times and of call arrival times. Finally, as shown in Figure 2.8, we can regard {N1 (t); 00}. By the same type of argument as in Section 2.3, the number of customers who have completed service by time ⌧ is independent of the number still in service. 2.5. CONDITIONAL ARRIVAL DENSITIES AND ORDER STATISTICS 2.5 Conditional arrival densities and order statistics A diverse range of problems involving Poisson processes are best tackled by conditioning on a given number n of arrivals in the interval (0, t], i.e., on the event N (t) = n. Because of the incremental view of the Poisson process as independent and stationary arrivals in each incremental interval of the time axis, we would guess that the arrivals should have some sort of uniform distribution given N (t) = n. More precisely, the following theorem shows that the joint density of S (n) = (S1 , S2 , . . . , Sn ) given N (t) = n is uniform over the region 0 < S1 < S2 < · · · < Sn < t. Theorem 2.5.1. Let fS(n) |N (t) (s(n) | n) be the joint density of S(n) conditional on N (t) = n. This density is constant over the region 0 < s1 < · · · < sn < t and has the value fS(n) |N (t) (s(n) | n) = n! . tn (2.37) Two proofs are given, each illustrative of useful techniques. Proof 1: Recall that the joint density of the first n + 1 arrivals S (n+1) = (S1 . . . , Sn , Sn+1 ) with no conditioning is given in (2.15). We first use Bayes law to calculate the joint density of S (n+1) conditional on N (t) = n. fS (n+1) |N (t) (s (n+1) | n) pN (t) (n) = pN (t)|S (n+1) (n|s (n+1) )fS (n+1) (s (n+1) ). Note that N (t) = n if and only if Sn  t and Sn+1 > t. Thus pN (t)|S (n+1) (n|s (n+1) ) is 1 if Sn  t and Sn+1 > t and is 0 otherwise. Restricting attention to the case N (t) = n, Sn  t and Sn+1 > t, fS (n+1) |N (t) (s (n+1) | n) = = = fS (n+1) (s (n+1) ) pN (t) (n) n+1 exp( t)n exp( ( n! exp( sn+1 ) t) /n! (sn+1 t) tn . (2.38) This is a useful expression, but we are interested in S (n) rather than S (n+1) . Thus we break up the left side of (2.38) as follows: fS (n+1) |N (t) (s (n+1) | n) = fS (n) |N (t) (s (n) | n) fSn+1 |S (n) N (t) (sn+1 |s (n) , n). Conditional on N (t) = n, Sn+1 is the first arrival epoch after t, which by the memoryless property is independent of S (n) . Thus that final term is simply exp( (sn+1 t)) for sn+1 > t. Substituting this into (2.38), the result is (2.37). Proof 2: This alternative proof derives (2.37) by looking at arrivals in very small increments of size (see Figure 2.9). For a given t and a given set of n times, 0 < s1 < · · · , < sn < t, we calculate the probability that there is a single arrival in each of the intervals (si , si + ], 1  i  n and no other arrivals in the interval (0, t]. Letting A( ) be this event, CHAPTER 2. POISSON PROCESSES - 0 - s1 - s2 s3 t Figure 2.9: Intervals for arrival density. Pr{A( )} = pN (s1 ) (0) pNe (s1 ,s1 + ) (1) pNe (s1 + ,s2 ) (0) pNe (s2 ,s2 + ) (1) · · · pNe (sn + ,t) (0). The sum of the lengths of the above intervals is t, so if we represent pNe (si ,si + ) (1) as exp( ) + o( ) for each i, then Pr{A( )} = ( )n exp( t) + n 1 o( ). The event A( ) can be characterized as the event that, first, N (t) = n and, second, that the n arrivals occur in (si , si + ] for 1  i  n. Thus we conclude that fS (n) |N (t) (s (n) ) = lim !0 Pr{A( )} , np N (t) (n) which simplifies to (2.37). The joint density of the interarrival intervals, X (n) = (X1 . . . , Xn ) given N (t) = n can be found directly from Theorem 2.5.1 simply by making the linear transformation X1 = S1 and Xi = Si S Pi 1 for 2  i  n. The density is unchanged, but the constraint region transforms into ni=1 Xi < t with Xi > 0 for 1  i  n (see Figure 2.10). @ @ @ @ x2 @ @ @ R @ @ @ s2 s1 x1 Figure 2.10: Mapping from arrival epochs to interarrival times. Note that incremental cubes in the arrival space map to parallelepipeds of the same volume in the interarrival space. fX (n) |N (t) (x (n) | n) = n! tn for X (n) > 0, Xn i=1 Xi < t. (2.39) 2.5. CONDITIONAL ARRIVAL DENSITIES AND ORDER STATISTICS It is also instructive to compare the joint distribution of S(n) conditional on N(t) = n with the joint distribution of n IID uniformly distributed random variables, U (n) = (U1, . . . , Un) on (0, t]. For any point U (n) = u(n), this joint density is fU (n) (u (n) ) = 1/tn for 0 < ui  t, 1  i  n. Both fS (n) and fU (n) are uniform over the volume of n-space where they are non-zero, but as illustrated in Figure 2.11 for n = 2, the volume for the latter is n! times larger than the volume for the former. To explain this more fully, we can define a set of random variables S1 , . . . , Sn , not as arrival epochs in a Poisson process, but rather as the order statistics function of the IID uniform variables U1 , . . . , Un ; that is S1 = min(U1 , . . . , Un ); S2 = 2nd smallest (U1 , . . . , Un ); etc. t pp pp p p pp pp pp pp pp pp p p pp pp pp p p p p p pp pp pp pp ppp p f =2/t2 p pp p s2 pp pp pp p p ppp ppp pp p pp p p p p p pp pp p p 0 0 s1 p p pp p pp t t pp pp pp pp pp pp pp pp pp pp pp pp pp pp pp pp pp p u2 pp pp pp 0 0 pp p p p pp pp pp p p p pp pp pp p p p pp pp pp pp pp pp pp pp pp pp pp pp pp pp pp pp pp p p p pp p p p ppp f =1/t2 p p p pp pp pp p p p p p p p p pp pp pp pp pp pp pp pp p p p pp pp pp pp pp pp pp pp pp pp pp pppppppp u1 p p p p p pp pp t Figure 2.11: Density for the order statistics of an IID 2-dimensional uniform distri­ bution. Note that the square over which fU (2) is non-zero contains one triangle where u2 > u1 and another of equal size where u1 > u2 . Each of these maps, by a permutation mapping, to the single triangle where s2 > s1 . The n-cube is partitioned into n! regions, one where u1 < u2 < · · · < un . For each permutation ⇡(i) of the integers 1 to n, there is another region10 where u⇡(1) < u⇡(2) < · · · < u⇡(n) . By symmetry, each of these regions has the same volume, which then must be 1/n! of the volume tn of the n-cube. All of these n! regions map to the same region of ordered values. Thus these order statistics have the same joint probability density function as the arrival epochs S1 , . . . , Sn conditional on N (t) = n. Anything we know (or can discover) about order statistics is valid for arrival epochs given N (t) = n and vice versa.11 Next we want to find the marginal distribution functions of the individual Si , conditional on N (t) = n. Starting with S1 , and viewing it as the minimum of the IID uniformly distributed 10 As usual, we are ignoring those points where ui = uj for some i, j, since the set of such points has 0 probability. 11 There is certainly also the intuitive notion, given n arrivals in (0, t], and given the stationary and independent increment properties of the Poisson process, that those n arrivals can be viewed as uniformly distributed. One way to view this is to visualize the Poisson process as the sum of a very large number k of independent processes of rate /k each. Then, given N (t) = n, with k >> n, there is negligible probability of more than one arrival from any one process, and for each of the n processes with arrivals, that arrival is uniformly distributed in (0, t]. CHAPTER 2. POISSON PROCESSES variables U1 , . . . , Un , we recognize that S1 > ⌧ if and only if Ui > ⌧ for all i, 1  i  n. Thus,  t ⌧ n Pr{S1 > ⌧ | N (t)=n} = for 0 < ⌧  t. (2.40) t For S2 to Sn , the density is slightly simpler in appearance than the distribution function. To find fSi |N (t) (si | n), look at n uniformly distributed rv’s in (0, t]. The probability that one of these lies in the interval (si , si + dt] is (n dt)/t. Out of the remaining n 1, the probability that i 1 lie in the interval (0, si ] is given by the binomial distribution with probability of success si /t. Thus the desired density is fSi |N (t) (x | n) dt = fSi |N (t) (si | n) = sii 1 (t si )n i (n 1)! tn 1 (n i)!(i 1)! n dt t sii 1 (t si )n i n! . tn (n i)!(i 1)! (2.41) It is easy to find the expected value of S1 conditional on N (t) = n by integrating the complementary distribution function in (2.40), getting E [S1 | N (t)=n] = t . n+1 (2.42) We come back later to find E [Si | N (t) = n] for 2  i  n. First, we look at the marginal distributions of the interarrival intervals. Recall from (2.39) that fX (n) |N (t) (x (n) | n) = n! tn for X (n) > 0, Xn i=1 Xi < t. (2.43) The joint density is the same for all points in the constraint region, and the constraint does not distinguish between X1 to Xn . Thus X1 , . . . , Xn must all have the same marginal distribution, and more generally the marginal distribution of any subset of the Xi can depend only on the size of the subset. We have found the distribution of S1 , which is the same as X1 , and thus  t ⌧ n Pr{Xi > ⌧ | N (t)=n} = for 1  i  n and 0 < ⌧  t. (2.44) t t E [Xi | N (t)=n] = for 1  i  n. (2.45) n+1 From this, we see immediately that for 1  i  n, E [Si | N (t) = n] = it n+1 (2.46) One could go on and derive joint distributions of all sorts at this point, but there is one additional type of interval that must be discussed. Define Xn⇤+1 = t Sn to be the interval from the largest arrival epoch before t to t itself. Rewriting (2.43), fX (n) |N (t) (x (n) | n) = n! tn for X (n) > 0, Xn⇤+1 > 0, Xn i=1 Xi + Xn⇤+1 = t. 2.6. SUMMARY The constraints above are symmetric in X1 , . . . , Xn , Xn⇤+1 , and, within the constraint re­ gion, the joint density of X1 , . . . , Xn (conditional on N (t) = n) is uniform. Note that there is no joint density over X1 , . . . , Xn , Xn⇤+1 condtional on n(t) = n, since Xn⇤+1 is then a de­ terministic function of X1 , . . . , Xn . However, the density over X1 , . . . , Xn can be replaced by a density over any other n rv’s out of X1 , . . . , Xn , Xn⇤+1 by a linear transformation with unit determinant. Thus Xn⇤+1 has the same marginal distribution as each of the Xi . This gives us a partial check on our work, since the interval (0, t] is divided into n + 1 intervals of sizes X1 , X2 , . . . , Xn , Xn⇤+1 , and each of these has a mean size t/(n + 1). We also see that the joint distribution function of any proper subset of X1 , X2 , . . . Xn , Xn⇤+1 is independent of the order of the variables. One important consequece of this is that we can look at a segment (0, t] of a Poisson process either forward or backward in time and it ‘looks the same.’ Looked at backwards, the interarrival intervals are Xn⇤+1 , Xn , . . . , X2 . These intervals are IID, and X1 is then determined as t Xn⇤+1 Xn · · · X2 . We will not make any particular use of this property here, but we will later explore this property of time-reversibility for other types of processes. For Poisson processes, this reversibility is intuitively obvious from the stationary and independent properties. It is less obvious how to express this condition by equations, but that is not really necessary at this point. 2.6 Summary We started the chapter with three equivalent definitions of a Poisson process—first as a renewal process with exponentially distributed inter-renewal intervals, second as a station­ ary and independent increment counting process with Poisson distributed arrivals in each interval, and third essentially as a limit of shrinking Bernoulli processes. We saw that each definition provided its own insights into the properties of the process. We emphasized the importance of the memoryless property of the exponential distribution, both as a useful tool in problem solving and as an underlying reason why the Poisson process is so simple. We next showed that the sum of independent Poisson processes is again a Poisson process. We also showed that if the arrivals in a Poisson process are independently routed to di↵erent locations with some fixed probability assignment, then the arrivals at these locations form independent Poisson processes. This ability to view independent Poisson processes either independently or as a splitting of a combined process is a powerful technique for finding almost trivial solutions to many problems. It was next shown that a non-homogeneous Poisson process could be viewed as a (ho­ mogeneous) Poisson process on a non-linear time scale. This allows all the properties of (homogeneous) Poisson properties to be applied directly to the non-homogeneous case. The simplest and most useful result from this is (2.34), showing that the number of arrivals in any interval has a Poisson PMF. This result was used to show that the number of cus­ tomers in service at any given time ⌧ in an M/G/1 queue has a Poisson PMF with a mean approaching times the expected service time in the limit as ⌧ ! 1. Finally we looked at the distribution of arrival epochs conditional on n arrivals in the interval (0, t]. It was found that these arrival epochs had the same joint distribution as the CHAPTER 2. POISSON PROCESSES order statistics of n uniform IID rv’s in (0, t]. By using symmetry and going back and forth between the uniform variables and the Poisson process arrivals, we found the distribution of the interarrival times, the arrival epochs, and various conditional distributions.