Chapr 4 Couous Tm Markov Chas Baba Goyal Ky words: Couous m sochasc procsss, Posso procss, brh procss, dah procss, gralzd brh-dah procss, succssv occurrcs, r-arrval m. Suggsd radgs:. Mdh, J. (996, Sochasc Procsss, Nw Ag Iraoal (P Ld.. Fllr, W.(968, A roduco o Probably Thory ad s Applcaos, Vol. I,Wly, Nw York, 3 rd do. 3. Karl, S. ad Taylor, H.M.(975, A frs cours Sochasc Procsss, Acadmc Prss. 4. Parz, E.(96, Iroduco o Sochasc Procsss, Uvrsal Book Sall. 5. Ross, S.M.(983, Sochasc Procsss, Joh Wly. 95
5. Iroduco Cosdr h followg procsss: ( A radoacv sysm ms a sram of α - parcls, whch rach a Ggr cour. A Ggr cour s a dfcv cour, whch gs lockd for somm wh a radoacv parcl srks. Whl a ulockd cour s capabl of rcordg h arrval of parcls, a lockd cour dos o rcord h v. L X ( b h umbr of rcordgs by a Ggr cour durg h m-rval (,]. Suppos ha h half-lf of h parcl (h m ak ll h mass of h mar rducs o s half by h procss of dsgrao s larg as compard o. Th X ( s h sum of a larg umbr of..d. Broull vs, ach havg a probably p (say of bg rcordd. ( A larg pod has a larg umbr of fshs. L X ( b h umbr of fshs caugh h mrval (,]. Th chac of cachg a fsh s dpd of h umbr of fshs alrady caugh, wh h cachg of a fsh s a Broull ral wh h probably p (say of cachg (succss. Furhr h chacs of cachg a fsh a h x m po ar h sam rrspcv of h m rval sc h las succss. I ohr words, hr s o prmum for wag. ( Rcall h quug sysm havg o srvr who srvs h cusomr o h bass of Frs com. Frs srvd rul. L X ( b h lgh of h quu a ay m, wh h cusomrs ar jog h sysm a a cosa ra ad hc h r-arrval ms ar..d. radom varabls. All h abov procsss rprs a suao wh h umbr of rals s larg, ad ach ral s subjc dpdly o a Broull law. Th probably of o occurrc of h v a vry small rval s cosa bu h sam rval wo or mor vs ca occur wh a probably, whch s of ordr zro. Ths all procsss ar h Posso procsss. 96
4. Posso procss L ( N(, T [, b a o-gav coug procss wh dscr sa spac S {,, } whr N ( dos h occurrc of a radom, rar v E m T. Furhr, l p ( P( N( P (of occurcs of E a rval (, ];,,... p (.., p ( s a fuco of m. Udr cra codos, calld as h posulas or assumpos, N(~Pos( λ. Th ( N (, T [, s calld a Posso procss. Posulas of Posso procss: Thr ar hr basc posulas or assumpos of h Posso procss: ( Idpdc: N( s Markova,.., h occurrcs of h v (,] ar dpd of h umbr of occurrcs of E a rval pror o (,]. ( Homogy m: p (dpds oly o h lgh of h m rval (,] ad o o whr h rval s suad o h m-axs. ( Rgulary (Ordrlss: I a rval (, + h of fsmal lgh h, xacly o v ca occur wh probably λ h+ o( h ad h probably of mor ha o vs s of ordr o( h (calld as h zro ordr,.., o( h p ( h λh + o( h ; lm h h ad, p ( h o( h k k Sc p ( h p ( h + p ( h + o( h p ( h λh o( h P(o v (, + h Udr hs posulas, w prov h followg horm: 97
Thorm 4.: Udr h codos of dpdc, homogy ad ordrlss, N( ~ Pos( λ,.., λ ( λ p (,,,... Proof: Cosdr p ( + h for + h For, p ( + h P( vs occur a rval (, + h P( vs occur h rval (, P( o vs occur (, + h + P( vs occur h rval (, P( o v occurs (, + h + P( k vs occur h rval (, P( k vs occur (, + h k p (( λh + p ( λh + o( h p ( + h p ( o( h λp( + λp ( h + h p ( + h p ( o( h + + h lm λp( λp ( lm h h h or, p '( + λp( λp ( ; (4. For, p ( + h p ( p ( h p (( λh + o( h p ( + h p ( lm λ p ( h h 98
p p '( ( λ or, log p ( λ+ c, c s h arbrary cosa of grao λ p ( c, c c Ially, a, p ( c Pug (4., w hav ( p λ (4. p '( + λp ( λp ( λ λ Igrag facor I.F. of hs dffral quao s s λ λ ( p ( d d λ p ( λ + c λ A, p ( c ( λ p λ L p ( λ ( λ (4.3 Th from (4. p '( + λp ( λ λ ( λ Mulplyg boh sds by λ, w hav λ ( p '( + λp ( ( λ λ 99
Igrag boh sds wh rspc o λ ( λ p ( + c p ( c λ ( λ p (,,,3,... Thus, by mahmacal duco, w hav show ha N (~Pos( λ,,,... Probably grag fuco ad h Characrsc fucos of a Posso procss: A alra ad mor lga chqu of obag h rsul s h grag fuco chqu. Df h probably grag fuco P( s, p ( s so, N( P( N ( s E( s P ( s, p ( s p ( + sp ( + s p ( +L p ( Dffrag P(s, parally w.r.., w hav P( s, p ( s p '( s p '( + p '( s (4.4
From (4., w hav p '( + λp( λp ( ; Mulplyg hs quao by s ad summg ovr all possbl valus of, w hav whr, + λ λ p '( s p ( s p ( s (4.5 p '( s P( s, p '( p ( s P(, s p ( (from (4.4 p ( s sp(, s so, (4.5 bcoms ( P( s, p '( λ P( s, p ( + λsp( s, P( s, λ( s P( s, ( p '( λp ( P( s, c λ ( s As P( s, ( s P( s, λ Hc, h p.g.f. of h procss s gv by λ( s λ P( s, ( λs p ( coffc of s P( s, λ ( λ,
N ( ~ Pos( λ,,,... Th characrsc fuco of hs procss s gv by sx ( ( φ ( s E λ s ( λs s λ( Dducos: ( E( N( λ Var( N( λ.., h ma ad h varac of N( dpd o ad as such, h procss s voluoary. ( I a rval of u lgh, h ma umbr of occurrcs s λ. Ths s calld h paramr of h procss. ( Posso procss s a couous paramr, dscr sa spac sochasc procss. Bu E (N ( s a couous o-radom fuco of. (v Posso procss has dpd ad saoary (m-homogous crms. (v If E occurrd r ms up o h al sa from whch s masurd h h al codo wll b p (, p ( r r p ( P( N ( r r λ ( λ, r r, < r (v As, for a Posso procss N (
P N( λ ε, whr ε > s a prassgd umbr. Usg Chbyshv's qualy for a radom varabl X, Pu X N ( Var( P( X E( X a X ; a > a λ P( N ( λ a ; a > a or, N( a λ P λ a L a ε N( P λ ε λ ε N( lm P λ ε.., for larg, N( ca b ak as a sma of λ. 4.3 Proprs of Posso procss ( Addv propry: Sum of wo dpd Posso procsss s aga a Posso procss. Proof: L N ( ad N ( b wo dpd Posso procsss wh paramrs λ ad λ rspcvly ad N( N ( + N (. Th P( N( P( N ( r, N ( r r P( N ( r P( N ( r du o dpdc r r r λ ( λ λ ( λ r r r 3
( λ + λ r λ λ ( ( r r r r + + ( λ + λ λ λλ λ λ +L ( λ + λ ( λ + λ, ( λ + λ (( λ+ λ, N (~Pos(( λ + λ ( ( Alravly, l h p.g.f. of N (,, b ( N E s λ s. Th p.g.f. of N( s N( N ( + N ( E( s E( s N ( N ( E ( s E ( s λ ( s λ ( s ( λ + λ ( s N (~Pos(( λ + λ Th c.f. of N ( s sn ( s ( N ( N ( ( ( + φ ( s E E E ( E ( s ( N ( s ( N ( s ( + λ ( λ ( Dffrc of wo dpd Posso procsss. Proof: L N ( ad N ( b wo dpd Posso procsss wh paramrs λ ad λ rspcvly ad N ( N ( N (. Th 4
P( N ( P( N ( + r, N ( r r + r r λ ( λ λ ( λ r + r r ( λ + λ λ λ λ + λ λ λ ( λλ + r r r + r ( I ( λλ [ ] whr, procss. I [ ] ( x r + r x r + r s modfd Bssl fuco of ordr ( -. N ( s o a Posso Alravly, h p.g.f. of N ( s N ( N ( N ( E( s E( s N ( N ( E ( s E ( s N ( E ( s E s λ λ ( s+ λ + λ s N ( Th p ( s h coffc of s h xpaso of N( E( s E( N( ( λ λ λ+ λ + λ λ E( N ( ( ( Var( N ( ( λ + λ Th c.f. of N ( s sn ( s ( N ( N ( ( ( φ ( s E E E ( E ( s ( N ( s ( N ( s s ( ( λ λ λ λ 5
( Dcomposo of a Posso procss: A radom slco from a Posso procss L N (, h umbr of occurrcs of a v E a rval of lgh s a Posso procss wh paramr λ. Furhr, l ach occurrc of E has a cosa probably p of bg rcordd, ad ha rcordg of a occurrc s dpd of ha of aohr occurrc ad of N (. If M ( s h umbr of occurrcs rcordd a rval of lgh, h M ( s also a Posso procss wh paramr λp. Proof: P( M ( P( E occurs ( + r ms by poch ad xacly ou of + r occurrcs ar rcordd r P( N ( + r P( vs ar rcordd r r + r λ r ( λ + r p q + r r λ ( λ p ( λq r r r λ ( λp λq λ( q ( λp λp ( λp M (~ Pos( λ p Th c.f. of M ( s s sm ( ( ( p φ ( s E λ Corollary: 6
. If M ( s h umbr of vs o bg rcordd, h M ( s a Posso procss wh paramr λ(-p λq. For xampl, a Ggr cour rcords radoacv dsgraos accordg o a Posso law. Also h dsgraos, whch hav o b rcordd, follow a Posso law.. If a Posso procss ca b brok up o r dpd srams wh probabls r p, p,... p r ; p, h, hs r dpd srams ar Posso procsss wh paramrs λ p λ p,... λ p rspcvly., r (v Posso procss ad Bomal dsrbuo: If N ( s a Posso procss h for s < s s P( N ( s k N ( k k k Proof: P( N ( s k N ( P( N( s k, N( P( N( P( N( s k, N( s k P( N( P( N( s k P( N( s k P( N( λs k λ( s k ( λs ( λ( s k k λ ( λ s ( s k k k k s s k k k (v If { N(, } s a Posso procss h h (auo corrlao coffc bw N ( ad N ( + s s + s. 7
Proof: L λ b h paramr of h procss, h E ( N ( T λt ; Var( N ( T λt E ( N ( T λt + ( λt ; T, + s. E( N( N( + s E( N( ( N( + s N( + N( E ( N ( T + E( N ( ( N ( + s N ( E ( N ( T + E( N ( E (( N ( + s N ( as N ( ad N ( + s ar dpd. λ+ λ + λ s Cov( N(, N( + s λ+ λ + λ s λλ ( + s λ λ Auocor( N (, N ( + s λλ ( + s + s I gral, ρ ( N(, N( ' m(, ' max(, ' Exampl ( M/G/ quu: Rcall h quug sysm whr cusomrs jo h sysm hopg for som srvc. Thr s a srvr who srvs o cusomr (f ay prs oly a m pos,, Th umbr of cusomrs Y h m rval (, +, ar..d. radom varabls.th srvc sao has a capacy of a mos c cusomrs cludg h o bg srvd ad furhr arrvals ar o rad by h srvc sao (los cusomrs. Furhr h srvc ms of succssv arrvals ar assumd o b dpd radom varabls wh a commo dsrbuo, say, G. ad hy ar dpd of furhr arrvals. Th {X, }, h umbr of cusomrs a m po s a Markov cha wh sa spac S {,, c}. W hav 8
Y, f X ad Y c X + X + Y, f X c ad Y c+ X c, ohrws If Y s a Posso procss, h j x ( x P( Y j λ λ dg( x ; j,,.. j ad h raso probabls of h Markov cha {X, ar j x ( x λ λ dg( x ;, j j j + λx ( λx p dg( x ; j, j j +, ohrws. 4.4 Posso dsrbuo ad rlad dsrbuos Takg a cu from h abov xampl, w ca ow dfy som dsrbuos, whch ar closly assocad wh h Posso procss. Ir-arrval m: L { N(, } s a Posso procss wh paramr λ. L X b h rval bw wo succssv occurrcs of h v E for whch N ( s h coug procss. Th X, kow as h r-arrval m s a radom varabl followg a xpoal dsrbuo. W sa ad prov h followg rsul. Thorm 4.: Th rval bw wo succssv occurrcs of a Posso procss{ N(, } wh paramr λ has a gav xpoal dsrbuo wh ma λ. 9
Proof: L X b h rval bw wo succssv occurrcs of { N(, }. L F ( x P( X x b h c.d.f. of X. X L E ad E + b h wo succssv occurrcs of h v E for whch N ( s h coug procss occurrg a m pochs ad + rspcvly. Th, P( X x P( E dd o occur (, E occurrd a sa > + + x P( E dd o occur (, N ( + + x P( N ( x N ( p ( x, x > λx Sc s arbrary, so for h rval X bw ay wo succssv occurrcs F ( x P( X x P( X> x X λx ; x> dfx ( x λx f ( x λ ; x> dx X ~xp( λ Th x rsul s a xso of hs rsul. Thorm 4.3: Th rvals bw succssv occurrcs of a Posso procss ar..d. xpoal varabls wh commo ma λ. Proof: W prov h rsul by mahmacal duco. L h succssv occurrc pos of h Posso procss ar < < <.... I h arlr horm, w hav provd ha h r-arrval m bw wo succssv occurrcs s a xpoal varabl wh ma,, ar h r-arrval ms, h λ. For hr succssv occurrcs, f X + -
P( X x, X > x P( E ddo occur (, E occurd a + + + + x + + x P( + > x + x X x f ( x dx P( > x + x X x P( N ( λx x λx λx P( X x, X > x λ dx λx λx ( L h rsul holds for k r-arrval ms X, X,..., X k. Th P( X x, X x, L X x, X > x k k k+ k+ x x x k k k k + λ x ( k+,, k k λ L P W > x X x X x LX x dx Ldxk whr k + k+ k P( Wk + > x X x, X x, LX k x k P N x N x P( N ( x k + k k k+ k+ λx k+ λx P( X x, X x, LX x, X > x ( L( λx k X, X,..., Xk ar..d. radom varabls. Th covrs of hs horm s qually ru, whch alog wh hs horm gvs a characrzao o h Posso procss. Thorm 4.4: If h rvals bw succssv occurrcs of a v E ar dpdly dsrbud xpoally wh commo ma, h h v E has Posso procss as s coug procss. λ
Proof: L { Z, } b a squc of..d. gav xpoal varabls wh commo ma λ, whr, h Z rval bw ( - ad occurrcs of h v E. h Df W Z + Z +... + Z as h wag m up o h h occurrc,.., h m from org o h h subsqu occurrc. Th, W ~ Gamma( λ, wh p.d.f. λ λ x x gw ( x!, x > ad c.d.f. F ( x P( W g ( x dx W Obvously, { N ( < } { W Z + Z +... + Z > },.. h wo c.d.f.'s F ad N ( FW sasfy h rlao F ( P( W P( W > W P( N ( < P( N ( F ( ( N F ( F ( N( W λ λx x dx! λ λx x dx! y y dy (pu λ x y! λ
j λ ( λ j! j (grao by pars p ( PN ( ( F ( F ( N( N( λ ( λ! N (~Pos( λ ; I may b od ha Posso procss has dpd xpoally dsrbud r-arrval ms ad Gamma dsrbud wag ms. Th x rsul xplas h purly radom aur of a Posso procss. Thorm 4.5: If a Posso procss N ( has occurrd oly oc by h m-po, h h dsrbuo of h m rval γ [,T], whch occurrd, s uform [,T],.., dt P( < γ + d N ( T ; < < T T Proof: W hav P( < γ + d N ( T λ d P( N ( T λt λ λt ad ( T P( N ( T γ λ s h probably ha hr was o occurrc of N ( h m rval (, T ]. Hc, P ( < γ + d N ( T P ( < γ + d, N ( T P ( N ( T P ( < γ + d P ( N ( T P ( N ( T λ d T λ λ ( T d. λ T λt 3
Th rsul ca b rprd as follows: If a Posso procss N ( has occurrd oly oc by h m-po, hs s qually lkly o happ aywhr [,T]. Ths s why h Posso procss s purly radom. W sa som mor rsuls, whch furhr mphasz h radom aur of Posso procss. ( For a Posso procss wh paramr λ, h m rval up o h frs occurrc also follows a xpoal dsrbuo wh ma λ,.., f X s h m up o h frs occurrc, h P( X > x P( N( x p ( x, x > λx.., P( X > x λx s dpd of ad. ( Suppos ha h rval X s masurd from a arbrary po ( r h rval + r > (, + ad o h po of h occurrc of E. L Y ( + r. Y s calld radom + modfcao of X or h rsdual m of X. Th, f X s xpoally dsrbud, so s s radom modfcao Y wh h sam ma. I ohr words, hr s o prmum for wag. ( Suppos ha A ad B ar wo dpd srs of Posso vs wh paramrs λ ad λ rspcvly. Df a radom varabl N as h umbr of occurrcs of A bw wo succssv occurrcs of B. Th N λ ~Go λ λ. + L X b h radom varabl dog h rval bw wo succssv occurrcs of B. Th x λ λ f ( x, x >. Hc, X 4
P( A occurs k ms a arbrary rval bw wo succssv occurrcs of B P( N k k ( λ λ k f ( d ( λ k d λ λ λ k λ λ k k ( λ + λ k d k λλ k + + λ ( λ λ λ ; k,,,... ( λ λ λ λ + + k (v Th abov propry ca b gralzd o df wha w call as a Posso cou procss. Posso cou procss: L E ad E' b wo radom squcs of vs occurrg a sas (,,... ad ( ', ',... rspcvly. Th umbr, N, of occurrcs of E' a rval (, s kow as h cou procss of E' E. If E s a Posso procss, h h cou procss s calld h Posso cou procss. If, alog wh E, E' s also a Posso procss h h cou procss N has a gomrc dsrbuo. N (, ar..d. gomrc varas. (v Suppos ha A ad B ar wo dpd srs of Posso vs wh paramrs λ ad λ rspcvly. Df a radom varabl N as h umbr of occurrcs of A bw wo succssv occurrcs of B. h rval bw wo coscuv occurrcs of B s h sum of wo dpd xpoal varas ad has h jo dsy f ( x λ x λ x 5
44443 444444443 λ ( x λx f x λ f ( x λ 44444444444444443 λ ( x f x λ x k λ ( λ λ λ k P( k occurrcs of A bw vry scod occurrc of B d k ( λ + λ k + k λ λ d k λ λ k+ k ( λ k + + λ k+ λ λ λ λ λ λ + + k ; k,,l.., h dsrbuo s gav bomal (covoluo of xpoal dsrbuo. 4.5 Gralzaos of Posso procss I h classcal Posso procss, s assumd ha h codoal probabls ar cosa,.., h probably of k vs h rval [, +h] gv occurrc of vs by m-po s gv by λh+ o( h, k p ( h P( N ( h k N ( o( h, k k λh+ o( h, k.., p k (h s dpd of as wll as. Ths procss ca b gralzd by cosdrg λ o mor a cosa bu a fuco of or or boh. Th gralzd procss s aga Markova aur. Ths gralzd procss has xcll rpraos rms of brh-dah procsss. Cosdr a populao of orgasms, whch rproduc o cra smlar orgasms. Th populao s dyamc as hr ar addos rms of brhs ad dlos rms of dahs. L b h sz of h populao a sa. Dpdg upo h aur of addos ad dlos h populao, varous yps of procsss ca b dfd. 6
4.5. Pur brh procss: L λ s a fuco of, h sz of h populao a sa. Th p( k, h, P( N( h k N( λh+ o( h, k o( h, k λ h+ o( h, k (4.6 Th, p ( + h p ( ( λ h + p ( λ h + o( h, p ( + h p ( ( λ h + o( h p '( λp( + λ p (, (4.7 p '( λ p ( (4.8 Ths s a pur brh procss (oly brhs ar hr ad o dahs as k s a o-gav gr. For spcfd al codos, a xplc xprsso for p ( ca b obad. Dpdg upo form of λ, dffr procsss ca b obad. ( Yul-Furry procss: L λ λ. Th (4.7 ad (4.8 ca b wr as p '( λp ( + ( λp (, p '( L h al codos b p ( ; p (,.., h procss sars wh oly o mmbr a m. Usg prcpl of mahmacal duco, w ow, oba a xprsso for p (. For, p '( λ p ( p ( c ; λ c s h cosa of grao. 7
A, p ( c p ( λ For, p '( λp ( + λp ( p '( + λp ( λ λ Igrag facor for hs quao s λ λ λ λ ( λ p d c + λ Sc p ( c p ( ( λ λ L λ λ p ( ( Now, p '( + λp ( ( λp ( λ λ ( λ ( Mulplyg boh sds by λ, w hav λ λ λ λ ( ( ( ( λ λ ( λ + p d c Sc p ( c λ λ p ( ( ; ad p ( { p (, }has gomrc dsrbuo wh paramr λ ad p.g.f. 8
λ λ P( s, ( s λ s λ s( E ( N ( P'( s, λ s λ Var( N( ( λ 4.5. Brh ad dah procss: Now, alog wh addos h populao, w cosdr dlos also,.., alog wh brhs, dahs ar also possbl. Df q( k, h, P(umbr of dahs (, + h k N( µ h+ o( h, k o( h, k µ h+ o( h, k (4.9 For k, µ (4.6 ad (4.9 oghr cosu a brh ad dah procss. Th probably of mor ha o brh or mor ha o dah s o (h. W wsh o oba p ( P( N( To oba h dffral-dffrc quao for p (, w cosdr h m rval (, + h (, + [, + h Sc, brhs ad dahs, boh ar possbl h populao, so h v { N ( + h, } ca occur h followg muually xclusv ways: E + j dvduals a m-po, brhs ad j dahs (, + h j, j,,... I s asy o s ha P( E, + j > o( h. Thrfor, j p ( + h P( E + P( E + P( E + P( E whr, 9
P( E P( o brh ad o dah (, + h N ( p ( ( λ h+ o( h( µ h+ o( h p ((( λ + µ h+ o( h P( E p ( ( λ h+ o( h( µ h+ o( h p (( λ h+ o( h P( E p ( ( λ h+ o( h( µ h+ o( h + + + p (( µ h+ o( h + + P( E p ( ( λ h+ o( h( µ h+ o( h p (( o( h o( h So for p ( + h p ( ( ( λ + µ h + p ( λ h + p ( µ h + o ( h + + p'( ( λ + µ p( + λ p ( + µ + p+ ( (4. For p( + h p( ( λ h+ o( h( µ h+ o( h + p( ( λ h+ o( h( µ h+ o( h p ( λ hp ( + µ hp ( + o( h λ + µ p '( p ( p ( (4. Ially a, f orgasms ar hr h populao, h p (, p ( (4. ad (4. rprs h dffral-dffrc quaos of a brh ad dah procss. W mak h followg assro:
For arbrary λ, µ, hr always xss a soluo p ( ( such ha p (. If λ ad µ ar boudd, h soluo s uqu ad sasfs p (. 4.5.3 Brhs ad dah ras: Dpdg upo h valus of λ ad µ, varous yps of brh ad dah procsss ca b dfd. ( Immgrao: Wh λ λ,.., λ s dpd of populao sz, h h cras h populao ca b rgardd as du o a xral sourc. Th procss s, h, kow as a mmgrao procss. ( Emgrao: Wh µ µ,.., µ s dpd of populao sz, h h dcras h populao ca b rgardd as du o lmao of som lms prs h populao. Th procss s, h, kow as a mgrao procss. ( Lar brh procss: Wh λ λ, h λ λh, s h codoal probably of o h brh a rval of lgh h, gv ha orgasms ar prs a h bgg of h rval. λ λ s h brh ra a u rval pr orgasm. λ. (v Lar dah procss: Wh µ µ, h h procss s kow as a lar dah procss. Wh h spcfc valus of boh λ ad µ ar cosdrd smulaously, w g h followg procsss: ( Immgrao-mgrao procss: Wh λ λ ad µ µ, h procss s kow as mmgrao-mgrao procss. Ths s a M/M/ quu. ( Lar growh procss: If for a brh ad dah procss P(a lm of h populao gvs brh o a w mmbr a small rval of lgh h λh + o( h
P( o brh rval (, + h N( λh+ o( h ad P( a lm of h populao ds a small rval of lgh h µ h+ o( h P( o dah a rval (, + h N( µ h+ o( h.., f for a brh ad dah procss, λ λ ad µ µ ( ; λ µ, h h procss s a lar growh procss. Ths procss, whch s voluoary aur, has xsv applcaos varous flds, parcularly, quug hory. Now w shall aalyz hs procss. Th dffral-dffrc quaos for hs procss ar p '( ( λ + µ p ( + ( λ p ( + ( + µ p (, + (4. ad p '( µ p ( (4.3 (a Grag fuco: L h p.g.f. of {p (} b P ( s, p ( s Th, P ( s, p ( s s ad P ( s, p '( s Mulplyg (4. by s, summg ovr,,3 ad h addg (4.3 o h rsul, w hav + λ + µ + λ + µ + + µ + p '( p '( s ( p ( s ( p ( s ( p ( p (
P P P P ( λ + µ s + λ s + µ s s s P ( µ ( λ + µ s + λs s Udr h al codo N (, h soluo of hs paral dffral-dffrc quao s gv by P( s, µ ( s ( µ λs λ ( s ( µ λs ( λ µ ( λ µ (4.4 Expadg P(s, as a powr srs, w g p (. (b Ma populao sz: Dffrag P(s, w.r.. s parally a s, w g h ma populao sz M( as M ( P( s, ( λ µ, f λ < µ As, M(, f λ > µ, f λ µ Sc hs mhod volvs dffrao of a o-so-asy p.g.f., so obag M( may b a b volvd xrcs. Alravly, M ( ca b obad from (4. ad (4.3 drcly. Now, M ( E ( N ( p ( Mulplyg boh sds of (4. by ad addg ovr dffr valus of, w hav λ + µ + λ + µ + + p '( ( p ( ( p ( ( p ( (4.5 whr, ( p ( ( p ( + ( p ( M ( + M ( whr, M ( E ( N ( p ( 3
ad, + + ( + p ( ( + p ( ( + p ( + ( M ( p ( ( M ( p ( M ( M ( ad, p '( M '( Thrfor, from (4.5, w g ( λ µ ( ( M '( ( λ + µ M ( + λ M ( + M( + µ M ( M( ( λ µ M ( M ( c, c bg h cosa of grao. Ially, ( M ( c M ( λ µ Aga, from (4. 3 λ + µ + λ + µ + + p '( ( p ( ( p ( ( p ( 3 ( M '( ( λ + µ M ( + λ ( + ( + ( p ( 3 3 + µ (( + ( + ( + p ( + M '( ( λ µ M ( + ( λ + µ M ( or, ( M '( ( λ µ M ( ( λ + µ λ µ ( λ µ ( λ + µ ( λ µ M ( + c λ µ Ially, M ( 4
( λ + µ c λ µ ( λ µ ( λ + µ ( λ µ ( λ + µ M ( + λ µ λ µ ( λ + µ ( λ µ ( λ µ Var( N ( (, f λ µ λ µ If λ µ, h M '( ( λ + µ M (.λ M (.λ+ c A, M ( M (.λ+ ad Var( N( λ (c Probably of xco: Sc λ, so s a absorbg sa,.. oc h populao rachs, rmas hrafr ad h populao bcoms xc. Whou ay loss of graly, l N (. Th (4.4 bcoms P( s, µ ( s ( µ λs λ ( s ( µ λs ( λ µ ( λ µ ( ( ( λ µ λ µ µ s( µ λ ( ( ( λ µ λ µ λ µ λs( a bs a bs c a c ds ds c whr, ( λ µ a µ ( b µ λ ( λ µ c d λ µ ( λ µ ( λ µ 5
So, a c ( λ µ µ ( P( N ( p ( ( λ µ λ µ P ( h populao wll vually bcom xc lm p ( ( λ µ µ ( lm ( λ µ λ µ µ <, f λ > µ λ, f λ > µ ad lm p ( for f λ < µ. Th physcal rprao of h probably of xco s ha f h brh ra s lss ha h dah ra a populao, h populao wll ulmaly bcom xc wh probably. If brh ra s mor ha h dah ra, h h populao bcoms xc wh probably lss ha uy. ( Lar growh wh mmgrao: For lar growh, λ ad oc h populao rachs, s boud o rma hr slf ad bcoms a absorbg sa. Howvr, f w assum ha alog wh brhs, addos h populao ar possbl hrough mmgraos also,.., som orgasms from som ohr populaos may also jo h populao udr cosdrao, h λ λ + a ( a > ad µ µ ( ; λ a, µ ad sa s o logr a absorbg sa. As soo as h brh ra rachs, som ohr orgasms jo h sysm ad h populao wll vr bcom xc (rflcg barrrs. Ths procss s h procss of lar growh wh mmgrao. (v Immgrao-dah procss: If λ λ ad µ µ (, h h brh ra s cosa ad dah ra s a lar fuco of, h h procss s kow as mmgrao-dah procss. Ths s h slf-srvc modl of h quug hory (M / M /. (v Pur dah procss: I hs cas, λ,.., o w brhs ohr ha hos prs a h bgg of h procss, ar possbl ad 6
P( of a dah (, + h µ h + o( h so, P( of a dah (, + h N ( µ h + o( h Ths procss s calld a pur dah procss. Now, for hs procss λ ad µ µ ( p ( + h p ( ( µ h + p ( (( + µ h + o ( h + p '( µ p( + ( + µ p + (, (4.6 ad, p ( + h p ( + p (( µ h + o( h p p '( µ ( (4.7 (4.6 ad (4.7 ar h dffral-dffrc quaos of a pur dah procss. To oba a xprsso for p (, w assum ha ally dvduals wr prs wh h procss bga. For, p '( µ p ( p '( p ( µ d or, (l p ( µ d µ p ( c, c bg h cosa of grao. Ially, p ( c p ( µ For -, p '( ( µ p ( + µ p (, p '( + ( µ p ( µ p ( 7
Igrag facor for h quao s ( µ d ( p ( d ( µ µ µ ( µ µ µ µ or, p ( d + c A, c p ( ( ( µ ( µ µ ( µ Procdg h smlar mar, w hav µ µ p ( ( ;,,..., Now, w procd o oba ma ad varac of a pur dah procss. Mulplyg (4.6 by ad summao ovr all possbl valus of, w hav whr, µ µ p '( p ( + ( + p ( ( µ M ( + µ M ( M ( + M ( E( N ( p ( ad, M ( E( N ( p ( M '( µ M ( M ( c µ Ially, M ( c M ( µ. Aga, from (4.6 8
3 µ + µ + + p '( p ( ( p ( 3 ( µ M ( + µ ( + ( + ( + p ( 3 + ( 3 µ M ( + µ M ( p ( M ( + p ( M ( + M ( 3 3 3 whr, M ( p ( M( µ M( + M( or, M ( + µ M ( µ µ µ µ M ( + c, whr c s h cosa of grao Now, M ( c µ µ M ( µ + ( µ µ µ Var( N ( ( Problms. A vsm ag slls shor-rm vsm polcs whch ar, gral, hr for o-yar or for wo-yars. H s abl o sll polcs accordg o a Posso procss wh ra pr moh wh wh probably.6, h slls o-yar polcs ad wh probably.4, h slls wo-yars polcs. If h rcvs a commsso % for o-yar polcs ad a commsso % for wo-yars polcs, fd hs xpcd ga as a prcag of busss do by hm. Wha s h varac of hs com?. A radoacv sourc ms parcls a a ra 5 pr mu accordac wh a Posso procss. Each md parcl has a probably.75 of bg rcordd. Fd h probably ha a -mu rval xacly rcordgs wll b hr. Wha s h probably of rcordg alas mssos durg h sam m-prod? 9
3. If N ( ad N ( ar wo dpd Posso procsss wh paramrs λ ad λ rspcvly, h show ha k P( N( k N( + N( p q k k whr p λ λ ; q λ + λ λ +λ 4. If, for a rval of vry small lgh h, o succss ca occur wh probably λh or o succss wh probably -λh, h h lmg dsrbuo wh h umbr of such (o-ovrlappg rvals ds o fy s a Posso procss. 5. Suppos ha cusomrs arrv a a cour accordac wh a Posso procss wh a ma ra of pr mu. Fd h probably ha h rval bw wo succssv arrvals s (a mor ha o mu (b lss ha 4 mus, ad (c bw ad 3 mus. 6. I a uclar physcs xprm, radoacv parcls from wo dffr sourcs hav b drcd o srk a a scr. Arrvals ar accordg o dpd Posso procsss wh ma ra λ pr mu from h frs sourc ad µ pr mu from h scod sourc. Th show ha h rval bw ay wo succssv arrvals has a gav xpoal dsrbuo wh ma pr mu. λ + µ 7. For a lar growh procss, fd h probably of ulma xco Wh h procss sars wh dvduals a m. 3