1 Notes on Little s Law (l = λw)

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1 Copyright c 29 by Karl Sigma Notes o Little s Law (l λw) We cosider here a famous ad very useful law i queueig theory called Little s Law, also kow as l λw, which asserts that the time average umber of customers i a queueig system, l, is equal to the rate at which customers arrive ad eter the system, λ, the average sojour time of a customer, w. For example, i a four-year college, i which (o average) 5 first-year studets eter per year, the average umber of studets preset at this college is give by 5 4 2,. Our presetatio is based o a sample-path aalysis ad the reader should ot assume apriori that ay specific stochastic assumptios are i force. Imagie istead that a sample path is beig studied of some stochastic queueig process.. Little s Law We cosider a queueig system i which customers arrive from the outside, sped some time i the system ad the depart. C deotes the th customer, ad this customer arrives ad eters the system at time t. The poit process {t : } is assumed a icreasig (to ) sequece of o-egative umbers with coutig process { : t }; max{ : t t} ( if there are o arrivals by time t), the umber of arrivals durig (, t]. Upo eterig the system, C speds W uits of time iside the system (C s sojour time) ad the departs the system at time t d t + W. Note that the departure times are ot ecessarily ordered, which meas that we do ot require that customers depart i the same order that they arrived (thik of a supermarket). {N d (t) : t } deotes the coutig process for the departure times {t d }; N d (t) the umber of customers who have departed by time t; ote that N d (t), t. A customer C is i the system at time t if ad oly if t t < t d t + W, ad we defie L(t), the total umber of customers i the system at time t, by () (2) (3) L(t) Defie (whe the its exist) I{t t < t d } I{W > t t } {:t t} I{W > t t }. (4) (5) λ w def, the arrival rate ito the system, def W j, average sojour time, Little s Law is amed after Joh D.C. Little, who was the first to prove a versio of it, i 96. Little s origial framework was stochastic however. I 974 S. Stidham proved a sample-path versio which is what we preset here.

2 (6) l def t L(s)ds, time average umber i system. Theorem. ( l λw) If both λ ad w exist ad are fiite, the l exists ad l λw. L λw is oe of the most geeral ad versatile laws i queueig theory, ad, if used i clever ways, ca lead to remarkably simple derivatios. The trick is to choose what the system is, ad what the arrivals to this system are. For example, give a complicated etwork of queues, the system ca be the waitig area of a isolated ode of iterest, or it ca be oe (or all together) of the service areas, etc. The area uder the path of L(s) from to t, L(s)ds, is simply the sum of whole ad partial sojour times (e.g., rectagles of height ad legths W j ). If the system is empty at time t, the the area is exactly W + + W ; otherwise some partial pieces must be cosidered. The followig iequality is easily derived: (7) To see this: (8) (9) () {j:t d j t} W j L(s)ds L(s)ds { {j:t j s t} W j I{W j > s t j }}ds t j I{W j > s t j }ds mi{w j, t t j }. Sice mi{w j, t t j } W j, the upper boud i (7) is immediate. For the lower boud () mi{w j, t t j } W j + t t j {j:t j +W j t} {j:t j t, t j +W j >t} (2) W j {j:t j +W j t} {j:t d j t} Dividig the upper boud by t, ad re-writig /t (/t)(/), we obtai ( ) t Takig the it as t yields λw, due to the assumed existece of the two ts i (4) ad (5) for λ ad w (ad their assumed fiiteess). Thus the proof of L λw ca be completed by showig that the lower boud i (7) whe divided by t coverges to λw as well, that is, we must show that (3) W j λw. {j:t d j t} 2

3 Lemma. If λ ad w exists ad are fiite, the (4) (5) W, W t. Proof : (6) (7) (8) W W j W j W j ( )( ) w w, W j by (5) ad fiiteess of w. (4) is thus proved. From (4) it follows that M(t )/t λ because it is assumed that t. Assumig that the arrival times are strictly icreasig yields M(t ) ad thus that M(t ) λ. t t If the arrival times are ot strictly icreasig (so-called batch arrivals), the Thus i either case, from (4) t M(t ) t λ. W W t t W M(t ) t λ, because λ is assumed fiite. (5) is thus proved. We are ow prepared to fiish the proof of L λw: Proof :[l λw] To prove (3) it suffices to prove (9) {j:t d j t} W j λw, because we already established λw as a upper boud. To this ed, choose ay ɛ > o matter how small. From Lemma. there exists a iteger such that W j ɛt j, j, ad thus that t d j t j + W j ( + ɛ)t j, j. 3

4 Thus from which it follows that {j : t d j t} {j : j, ( + ɛ)t j t} {j : j, t j t + ɛ }, The rhs of the above ca be re-writte as {j:t d j t} W j N( t +ɛ ) N( t +ɛ ) W j j Dividig the first piece by t ad lettig t yields λw/( + ɛ) by the same argumet used o the upper boud i (7). The secod piece is a costat hece whe divided by t, teds to. Thus we coclude that for ay ɛ >, {j:t d j t} W j λw/( + ɛ). Sice ɛ > was chose arbitrary, we coclude that (9) holds. A cosequece of the proof of Theorem. (l λw) is Propositio. If λ exists ad is fiite, ad if W /, the N d (t) λ, the departure rate exists ad equals the arrival rate λ: Departure rate arrival rate. Proof : Sice N d N (t), d (t) λ; a upper boud is established. Now for a lower boud: (5) followed from (4) oly (a coditio that is weaker tha assumig w exists ad is fiite); hece as i the proof of l λw, for every ɛ > there exists a iteger such that N d (t) N(t/( + ɛ)), yieldig N d (t) λ..2 Applicatios of l λw. Q λd: If we let the system be the queue area (where customers wait before eterig service), the average sojour time is average delay i queue, d, l becomes average umber waitig i queue, Q, ad l λw takes o the form Q λd. 4

5 2. Ifiite server queue: For ay ifiite server queue with arrival rate λ < ad average service time /µ <, l exists ad l ρ λ/µ, because w /µ here: W S. 3. Proportio of time the server is busy i a sigle-server queue: Customers arrive to the queue at rate λ < ad have average service time /µ <. Let λ s deote the rate at which customers eter service. Lettig the system be the server, ad lettig L s (t) deote the umber of customers i service at time t, with time-average l s, we coclude that l s λ s (/µ), because W S here. It ca be proved that λ s λ whe ρ < ad λ s µ whe ρ. Thus l s ρ if ρ < ; l s if ρ. But sice L s (t) if the server is busy at time t, ad L s (t) if the server is idle at time t, we coclude (from the fact that l s is a time average) that l s is the log-ru proportio of time the server is busy: For ay sigle-server queue, with arrival rate λ ad mea service time /µ, the log-ru proportio of time the server is busy exists ad is equal to mi{ρ, }. 4. FIFO M/M/ queue with ρ < : We earlier solved for the balace equatios of umber i system; P ( ρ)ρ,. Thus l P ρ/( ρ). l λw implies that w l/λ yieldig the followig expressio for average sojour time: w µ ρ. This has a ice iterpretatio: From PASTA, the average umber of customers foud i the system by a arrival is the same as the time average umber i system, l. Moreover, by the memoryless property of the service times, the customer i service has a remaiig service as good as ew. Thus (addig i their ow service time), such a arrivig customer must wait, o average, for the completio of + l iid service times each of mea /µ. Thus w (/µ)( + l) µ ρ. 5

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