1 Notes on Little s Law (l = λw)

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1 Copyrigh c 26 by Karl Sigma Noes o Lile s Law (l λw) We cosider here a famous ad very useful law i queueig heory called Lile s Law, also kow as l λw, which assers ha he ime average umber of cusomers i a queueig sysem, l, is equal o he rae a which cusomers arrive, λ, he average sojour ime of a cusomer, w. For example, i a four-year college, i which (o average) 5 firs-year sudes eer per year, he average umber of sudes prese a his college is give by 5 4 2,. Afer preseig l λw, we offer, i he same spiri, a more geeral law kow as H λg ha allows oe o aalyze differe queueig quaiies of ieres besides umber i sysem, bu is based o he same elemeary priciples ad mehods. Our preseaio is based o a sample-pah aalysis ad he reader should o assume apriori ha ay specific sochasic assumpios are i force. Imagie isead ha a sample pah is beig sudied of some sochasic queueig process.. Lile s Law We cosider a queueig sysem i which cusomers arrive from he ouside, sped some ime i he sysem ad he depar. C deoes he h cusomer, ad his cusomer arrives ad eers he sysem a ime. The poi process { : } is assumed a icreasig (o ) sequece of o-egaive umbers wih couig process { : }; max{ : } ( if here are o arrivals by ime ), he umber of arrivals durig (, ]. Upo eerig he sysem, C speds uis of ime iside he sysem (C s sojour ime) ad he depars he sysem a ime d +. Noe ha he deparure imes are o ecessarily ordered, which meas ha we do o require ha cusomers depar i he same order ha hey arrived (hik of a supermarke). {N d () : } deoes he couig process for he deparure imes { d }; N d () he umber of cusomers who have depared by ime ; oe ha N d (),. A cusomer C is i he sysem a ime if ad oly if < d +, ad we defie L(), he oal umber of cusomers i he sysem a ime, by () (2) (3) L() I{ < d } I{ > } {: } I{ > }, where I{A} deoes he idicaor fucio for he eve A: I{A} if A occurs; oherwise. Defie (whe he is exis) (4) λ def, he arrival rae io he sysem, Lile s Law is amed afer Joh D.C. Lile, who was he firs o prove a versio of i, i 96. Lile s origial framework was sochasic however. I 974 S. Sidham proved a sample-pah versio which is wha we prese here.

2 (5) (6) w l def W j, average sojour ime, j def L(s)ds, ime average umber i sysem. Theorem. ( l λw) If boh λ ad w exis ad are fiie, he l exiss ad l λw. l λw is oe of he mos geeral ad versaile laws i queueig heory, ad, if used i clever ways, ca lead o remarkably simple derivaios. The rick is o choose wha he sysem is, ad wha he arrivals o his sysem are. For example, give a complicaed ework of queues, he sysem ca be he waiig area of a isolaed ode of ieres, or i ca be oe (or all ogeher) of he service areas, ec. The area uder he pah of L(s) from o, L(s)ds, is simply he sum of whole ad parial sojour imes (e.g., recagles of heigh ad leghs W j ). If he sysem is empy a ime, he he area is exacly W + + W ; oherwise some parial pieces mus be cosidered. The followig iequaliy is easily derived: (7) To see his: (8) (9) () {j: d j } W j L(s)ds L(s)ds { {j: j s } W j j I{W j > s j }}ds j I{W j > s j }ds mi{w j, j }. Sice mi{w j, j } W j, he upper boud i (7) is immediae. For he lower boud () mi{w j, j } W j + j (2) {j: j +W j } {j: j +W j } W j {j: j, j +W j >} {j: d j } Dividig he upper boud by, ad re-wriig / (/)(/), we obai ( ) j Takig he i as yields λw, due o he assumed exisece of he wo s i (4) ad (5) for λ ad w (ad heir assumed fiieess). Thus he proof of l λw ca be compleed by showig ha he lower boud i (7) whe divided by coverges o λw as well, ha is, we mus show ha (3) W j λw. {j: d j } 2

3 Lemma. If λ ad w exiss ad are fiie, he (4) (5) Proof :,. (6) (7) (8) W j W j j j W j ( )( ) j j w w, W j by (5) ad fiieess of w. (4) is hus proved. From (4) i follows ha N( )/ λ because i is assumed ha. Assumig ha he arrival imes are sricly icreasig yields N( ) ad hus ha N( ) λ. If he arrival imes are o sricly icreasig (so-called bach arrivals), he Thus i eiher case, from (4) N( ) λ. N( ) λ, because λ is assumed fiie. (5) is hus proved. We are ow prepared o fiish he proof of l λw: Proof :[l λw] To prove (3) i suffices o prove (9) {j: d j } W j λw, because we already esablished λw as a upper boud. To his ed, choose ay ɛ > o maer how small. From Lemma. here exiss a ieger m such ha W j ɛ j, j m, ad hus ha d j j + W j ( + ɛ) j, j m. Thus {j : d j } {j : j m, ( + ɛ) j } {j : j m, j + ɛ }, 3

4 from which i follows ha The rhs of he above ca be re-wrie as {j: d j } W j N( +ɛ ) j N( +ɛ ) jm m W j Dividig he firs piece by ad leig yields λw/( + ɛ) by he same argume used o he upper boud i (7). The secod piece is a cosa hece whe divided by, eds o. Thus we coclude ha for ay ɛ >, j {j: d j } W j λw/( + ɛ). Sice ɛ > was chose arbirary, we coclude ha (9) holds via leig ɛ. A cosequece of he proof of Theorem. (l λw) is Proposiio. If λ exiss ad is fiie, ad if /, he N d () λ, he deparure rae exiss ad equals he arrival rae λ: Deparure rae arrival rae. Proof : (5) followed from (4) oly (a codiio ha is weaker ha assumig w exiss ad is fiie); hece as i he proof of l λw, for every ɛ > here exiss a ieger m such ha N d () N(/( + ɛ)) m, yieldig N d () λ. Sice N d N (), d () he resul. λ; he upper boud holds as well yieldig.2 Applicaios of l λw. Q λd: If we le he sysem be he queue area (where cusomers wai before eerig service), he average sojour ime is average delay i queue, d, l becomes average umber waiig i queue, Q, ad l λw akes o he form Q λd. 2. Ifiie server queue: For ay ifiie server queue wih arrival rae λ < ad average service ime /µ <, l exiss ad l ρ λ/µ, because w /µ here: S. 4

5 3. Proporio of ime he server is busy i a sigle-server queue: Cusomers arrive o he queue a rae λ < ad have average service ime /µ <. Le λ s deoe he rae a which cusomers eer service. Leig he sysem be he server, ad leig L s () deoe he umber of cusomers i service a ime, wih ime-average l s, we coclude ha l s λ s (/µ), because S here. I ca be proved ha λ s λ whe ρ < ad λ s µ whe ρ. Thus l s ρ if ρ < ; l s if ρ. Bu sice L s () if he server is busy a ime, ad L s () if he server is idle a ime, we coclude (from he fac ha l s is a ime average) ha l s is i fac he log ru proporio of ime he server is busy: The log-ru proporio of ime he server is busy i a sigle-server queue mi{, ρ}. 4. Average umber of busy servers i a c server queue: For ay c server queue i parallel (wih oe lie o wai i) wih arrival rae λ < ad average service ime /µ <, le he sysem be he collecio of c servers (e.g., we are leavig ou he lie). Le ρ λ/µ, ad assume ha ρ < c. The l for his sysem is he average umber of cusomers i service, which equivalely is he average umber of busy servers; i exiss ad l ρ, because w /µ here: S. 5

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