(average number of points per unit length). Note that Equation (9B1) does not depend on the

Size: px

Start display at page:

Download "(average number of points per unit length). Note that Equation (9B1) does not depend on the"

Roland Harper
6 years ago
Views:

1 EE603 Class Notes 9/25/203 Joh Stesby Appeix 9-B: Raom Poisso Poits As iscusse i Chapter, let (t,t 2 ) eote the umber of Poisso raom poits i the iterval (t, t 2 ]. The quatity (t, t 2 ) is a o-egative-iteger-value raom variable with k ( ) P[ (t,t 2) k] p k( ) e, k 0, (9B) k! where t - t 2 is the iterval legth, a costat > 0 is the average poit esity (average umber of poits per uit legth). Note that Equatio (9B) oes ot epe o the absolute value of t or t 2. We say that (t, t 2 ) is Poisso istribute with parameter Raom Poisso poits have iepeet icremets; the umber of poits i (t, t 2 ] is iepeet of the umber of poits i (t 3, t 4 ] if these two itervals o ot overlap (i.e., the itersectio of these itervals is the ull set). This completes our first characterizatio of raom Poisso poits. I what follows, two other (equivalet) characterizatios are iscusse. From the results of Chapter 2 of the class otes, i a iterval of legth, the expecte umber of poits is. Somewhat surprisigly, this is also equal to the variace of (t, t 2 ); we write E[ (t,t )] VAR[ (t,t )]. (9B2) 2 2 A seco characterizatio of raom Poisso poits has to o with the uratio betwee poits. As show below, this uratio is a raom variable that is expoetially istribute. Raom Poisso poits ca be characterize by their iepeet-icremet ature as well as a requiremet that the uratio betwee poits is escribe by a expoetial raom variable (this is our seco characterizatio). Fially, a thir characterizatio ca be give for raom Poisso poits. Raom Poisso poits ca be characterize as a iepeet-icremet poit process that obeys the Upates at 9B-

2 EE603 Class Notes 9/25/203 Joh Stesby Markov property (or memory-less property) that was iscusse towars the e of Chapter 2 of these class otes (see Examples (2-4) a (2-5)). These alterative characterizatios are iscusse i this appeix. Give the three characterizatios metioe above, it shoul be apparet that raom Poisso poits ca be use to moel may physical pheomea. They serve as a goo moel i may situatios where somethig occurs repeately, the repetitios are iepeet of each other, a the average umber of repetitios per uit time (or uit legth) is costat over a large umber of successive uit-time (or uit-legth) itervals. Applicatios that ivolve N poits place raomly i a iterval of legth T ca, whe N a T are large, be moele usig Poisso poits. As iscusse i Chapter of the class otes, a Poisso poit moel becomes exact as N a T approach ifiity i such a maer that the ratio N/T approaches a costat (the average umber of poits per uit legth). Poisso poit moels have bee applie to problems ealig with electro emissio i semicouctors a vacuum tubes (i.e., shot oise i electroic evices), telephoe calls arrivig at a switchboar a the arrival of cars at a itersectio, to ame a few applicatios. Expoetially Distribute Duratio The uratio betwee ajacet Poisso poits is iepeet a expoetially istribute. First, we show a simpler result: we aalyze the statistics of the uratio from a arbitrary, but fixe, value of time to the earest Poisso poit. Let t be ay fixe marker/referece poit i time; we show that the uratio from t to the earest Poisso poit (o either sie of t ) is expoetially istribute. Defie raom variable to be the uratio from a fixe, but arbitrary, marker/referece poit t 0 to the first Poisso poit to the right of t 0 (see Fig. 9B-). For algebraic variable, the evet [ ] is equivalet to the evet there are oe or more Poisso poits i the iterval (t 0, t 0 +]. That is, [ ] = [(t 0, t 0 +) ] so that Upates at 9B-2

3 EE603 Class Notes 9/25/203 Joh Stesby - - T t 0 Deotes Locatio of a Poisso Poit t 0 is a arbitrary fixe poit 0 ( ) 0! [ ] [ (, ) ] [ (, ) 0] e e t t t t (9B3) P P P Fig. 9B-: Poisso poits (a Poisso poit process). We have - F ( ) -e, 0, (9B4) a, - f ( ) e, 0, (9B5) for the istributio a esity fuctios, respectively, that escribe raom variable. Equatio (9B5) establishes the esire result that raom variable is expoetially istribute. Now, let raom variable - eote the uratio from t 0 back to the first Poisso poit before t 0 (see Fig 9B-). I a maer similar to that use above, oe ca show that is expoetially istribute. The statistical properties of the uratio from a arbitrary fixe poit t 0 to ay Poisso poit (to the right or left of t 0 ) ca be aalyze. Defie raom variable as the uratio from t 0 to the th Poisso poit to the right of t 0. The evet [ ] is equivalet to the evet there are or more Poisso poits i the iterval (t 0, t 0 + ] so that Upates at 9B-3

4 EE603 Class Notes 9/25/203 Joh Stesby - k ( ) k! k=0 P P P [ ]= [ ( t, t + ) ] = - [ ( t, t + ) ] = - e. (9B6) We have - k ( ) - F ( ) - e, (9B7) k=0 k! the istributio fuctio for. Fially, ifferetiate (9B7) to obtai the gamma esity fuctio k k- k 2 k ( ) ( ) ( ) ( ) f ( ) e e k! (k -)! k! k! k0 k k0 k0 - - e, 0. ( -)! (9B8) (some authors refer to (9B8) as a Erlag esity see p. 459 of Stark a Woos, 4 th eitio). This last result allows us to aalyze the statistical properties of the uratio betwee Poisso poits. As show by Figure 9B-, efie raom variable T - -, the uratio betwee the (-) th a the th Poisso poits to the right of arbitrary fixe poit t 0. Sice Poisso poits i o-overlappig itervals are iepeet, the raom variables T a - are iepeet, a the esity fuctio that escribes is the covolutio of the esities that escribe T a - (ca you explai why this is true?). Equivaletly, the momet geeratig fuctio (s) for is equal to the prouct T (s) (s) of momet geeratig fuctios for T a -, respectively (agai, ca you explai why this is true?). So, usig (9B8) a the efiitio of the momet geeratig fuctio, we calculate sx - - x sx - -( s)x 0 0 τ (s) f (x)e x = x e e x = x e x. (9B9) ( -)! ( -)! Upates at 9B-4

5 EE603 Class Notes 9/25/203 Joh Stesby The last itegral o the right of (9B9) is tabulate as () ( )!, (9B0) - - ( s)x x e x = 0 ( s) ( s) where () = (-)! is the well kow Gamma fuctio with iteger argumet. Fially, substitute (9B0) ito (9B9) to obtai () (s) =. (9B) ( -)! ( s) ( s) Fially, sice (s) = T (s) (s), we have (s). (9B2) (s) ( s) T (s) ( s) ( s) Note that T (s) is the momet geeratig fuctio of a expoetial raom variable. Hece, raom variable T, the uratio betwee the (-) th a th Poisso poits to the right of t 0 (or ay two Poisso poits), is escribe by a expoetial raom variable with parameter. Poisso Poits Obey the Markov Property Poisso poits obey the Markov property (see the System Reliability sectio i Chapter 2). As above, eote t 0 as a fixe (but arbitrary) marker/referece poit. Deote as a raom variable that escribes the time uratio betwee t 0 a the ext Poisso poit to the right, raom variable beig escribe by esity f ( ). Let coitioal esity f( > ) escribes raom variable coitioe o the evet [ > ], a algebraic variable. Sice f ( ) is expoetial, we have the Markov property Upates at 9B-5

6 EE603 Class Notes 9/25/203 Joh Stesby f( > ) = f ( ), 0, (9B3) as show i Chapter 2 (see Examples (2-4) a (2-5)). A similar formula ca be writte for the (coitioal) esity that escribes raom variable T coitioe o the evet T >. The fact that it has bee very log (or short) secos sice t 0 a ot observig a poit oes ot make the ext Poisso poit more (or less) likely. State ifferetly, the probability of fiig a poit i the iterval (, ] is ot ifluece by the legth of (t 0, ], a iterval cotaiig o poit(s). Developmet of (9B3) Recall that eotes the istace from ay marker/referece t 0 to the first Poisso poit after t 0. It is expoetially istribute with parameter > 0. The istributio of coitioe o the evet > is P[, ] P[ < ] F ( ) F ( ) P P (9B4) F( ), 0 [ ] - [ ] F ( ) 0, otherwise Distributio fuctios F () a F( ) are right cotiuous. As a result, as, F( ) F( ) = 0The esity of coitioe o the evet > is f ( ) f( ) F( ), 0 F ( ) 0, otherwise (9B5) Note that f( ) as. Recall that is expoetially istribute; the istributio (9B4) a esity (9B5) are repeate here as Upates at 9B-6

7 EE603 Class Notes 9/25/203 Joh Stesby - F ( ) -e, 0, (9B6) a, - f ( ) e, 0, (9B7) respectively. Substitute (9B7) ito (9B5) to obtai e f( ) 0 e ( ) e (9B8) f ( ) 0. Startig from ay arbitrary marker/referece poit t 0, the fact that it has bee very log (or short) secos sice t 0 a ot observig a poit oes ot make the ext Poisso poit more (or less) likely. State ifferetly, the fact that o poit is i (t 0, ], > t 0, oes ot ifluece what will be fou i (, ],. State agai, the fact that o poit is i (t 0, ], > t 0, is forgotte whe it comes to characterizig the poits i (, ],, a attribute kow as the memory-less, or Markov, property. We have just show that expoetial raom variable has the Markov property (as o all expoetial raom variables). The coverse is true as well. If raom variable has the Markov property, it must be expoetially istribute. To show this, start by assumig that has the Markov property (9B3), a rea agai the setece followig (9B5) to coclue limit f ( ) F ( ) (9B9) Upates at 9B-7

8 EE603 Class Notes 9/25/203 Joh Stesby for some costat > 0. However, this last result implies F ( ) limit F ( ). (9B20) Sice F (0) 0, this last ifferetial equatio implies that F ( ) e, 0, so that is expoetially istribute. Iepeet icremets, a average of poits per uit legth, a the Markov property are three attributes that imply/characterize Poisso poits. Upates at 9B-8

Inhomogeneous Poisson process

Inhomogeneous Poisson process Chapter 22 Ihomogeeous Poisso process We coclue our stuy of Poisso processes with the case of o-statioary rates. Let us cosier a arrival rate, λ(t), that with time, but oe that is still Markovia. That