The output of a Queuing System

The output of a Queuing System - THE OUTPUT OF A QUEUING...

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Unformatted text preview: THE OUTPUT OF A QUEUING SYSTEM PAUL J. BURKE Bell Telephone Laboratorzes, New York, New York (Received June 25, 1956) For a queulng system wrth Poxsson Input, 3. sxngle waltlng lme thhout defections, and Identlcally dxstrxbuted Independent (negatlve) exponen- txal semce txmes, the equlhbnum dxstnbutlon of the number of semce completions m an arbitrary tune mterval 1s shown to be the same as the Input distrlbutlon, for any number of servers Thls result has applxca- tions In problems of tandem queulng The essence of the proof Is the demonstration of the mdependence of an Interdeparture Interval and the state of the system at the end of the Interval HE PROBLEM OF the output or ‘efllux,’ as termed by MOBSE,m of a queumg system apparently has been prevmusly consrdered, but not mvestlgated, 1n the hterature Thus Morse m the reference (31th states, “A httle thought W111 convmce one that the efilux from a smgle-channel, exponentlal semce channel, fed by Poxsson amvals, must be Poxsson w1th the same rate as the amvals ” In s1mxlar vem, for the case of two gates 1n senes wrth amvals and serv- lce tlmes random at the first gate, O’BRIEN has stated, “The amval of customers at gate 2 w111 be random and the average arnval rate W111 be exactly the same as that for gate 1 . . ” [2‘ Nelther statement 18 supported by analysus Smce the truth of these statements 1s far from obv10us, a proof seems necessary It Is the mam purpose of this paper to supply that proof and, although the quoted state- ments apply only to smgle—server queues, to generahze the result to multl— server queues.* The motlvatlon for studying the output dlstnbutlon anses chlefly m problems of tandem queumg, whlch occur m a vanety of apphcatlons One example 1s that of customers 111 a store who must first be wasted on by sales clerks and then, after bemg served by these clerks, must then be served by wrappers or cashlers Another, more comphcated example, 18 the settmg up of a telephone call through a smtchmg system. One type of the latter problem has been studred by V E BENES m an unpubhshed memorandum Th1s problem of Benes’ 1s concerned w1th a two-stage queue, the first or reglster stage havmg random mput, exponentlal servme trmes, and an arbltrary number of servers, wh11e the second or marker stage has one server and general semce tlme The prmclpal result 1s that the equl- hbnum dlstnbutlon of queue length and elapsed semce tune 1n the second * A dlflerent method of proof which meludes the present result as a speclal case was developed, after the completlon of this paper, by DR EDGAR REICH of the Rand Corporation Dr Relch’s method wxll be published m the Annals of Mathematical Slammer 699 700 Paul J. Burke stage 15 the same as though the Input to the second stage were random In Benes’ words, “If we had consrdered the marker.system by Itself w1th P01sson mput at rate a calls per second, we would have obtamed preemer the equatlon for the generatmg functlon of the number of people present and the age of the marker ” Thus 1f 1t had been known that the output from the first stage was 1n fact random, the result of thls paper would have followed from well-known theorems w1th a cons1derable savmg 1n analys1s Stlll another example of tandem queumg 1s analyzed 1n a paper by R R P JACKSON [31 In one case of Jackson’s problem, random mput to the first stage 1s assumed, and each stage has a smgle server w1th exponentlal serv1ce t1me Agam, certaln of Jackson’s results follow nnmed1ately, 1f 1t 1s known that the output of each stage has the same Porsson d1str1butlon as the or1g1nal Input, and, furthermore, these results can be generallzed to any number of stages It Is mtultlvely clear that, 1n tandem queumg processes of the type men- tloned above, If the output dlstrrbutlon of each stage was of such character that the queumg system formed by the second stage was amenable to analy- sxs, then the tandem queue could be analyzed stage—by-stage msofar as the separate delay and queue-length d1str1but1ons are concerned Such a stage—by—stage analys1s can be expected to be c0ns1derably snnpler than the srmultaneous analys1s heretofore necessary Fortunately, under the condxtlons stated below, 1t 15 true that the output has the requ1red s1m- p11c1ty for treatmg each stage md1v1dually THE MODEL A SUMMARY statement of the theorem proved 1n th1s paper 1s that the out- put of a queumg system w1th P01sson Input and exponentlal holdmg tunes 1s agam P01sson The detalls of the hypothes1s are as follows A smgle—stage queue w1th random 1nput 1s assumed The average mterarrlval mterval has length 1 / A That 1s, the probabrllty of the arrrval of a call (customer) dunng an mterval of length dt 1s taken equal to >\ dt, w1thm mfimtesrmals of h1gher order, 1ndependent of the state of the system, arrrval tlmes of pre- v10us calls, or any other condltlons Whatever There are 8 servers (channels) each havmg an exponentlal holdmg tune d1str1but1on w1th average 1/11 The holdmg tunes are completely mde- pendent of all condltlons Hence the probablhty that a call Wluch 1s re— celvmg serv1ce at the begmmng of an mterval dt Wlll terrmnate durmg the mterval rs [.1 dt W1thm lnfimtesrmals of hlgher order Under these assumpt1ons and the further cond1t10n su>)\, 1t 13 well known that there 1s an ethbrlum d1str1but1on of the states (number of calls 1n the system) Furthermore th1s d1str1but10n 1s the same as that of the states encountered by calls enterlng the system Output of a Queuing System 701 All calls remam 1n the system unt11 they have rece1ved serv1ce Other- w1se, the queue d1sc1pl1ne, or order of serv1ce, 1s 1rre1evant, smce the output and not the delay dlstrlbutlon 1s of 1nterest OUTLINE OF PROOF IN ORDER TO SHOW that the equlhbrlum d1str1but1on of the number of calls completmg serv1ce durmg an arb1trary mterval of length T 1s P01sson \Vlth parameter KT under the cond1t1ons of the model, an equu alent result, that the t1me mtervals between success1ve call complet1ons are 1ndependently dlstrlbuted WIth the same exponentlal d1str1but1on as the t1me mtervals between arrlvals, 11 111 be obtamed meg to the randomness of the Input and to the exponent1al holdmg- t1me dlstnbutmn, the output process rs Markoffian w1th respect to the state of the system, 1 e , glven the state of the system at any t1me t, no further knowledge concernlng the output d1str1but10n subsequent to t 15 gamed from the prev1ous h1story of the system It w111 be shown that an mter- departure mterval and the state of the system at the end of the mterval are 1ndependent at equ111br1um Together W1th the Markolfian property, thls 1ndependence lmphes that all mterdeparture mtervals are mdependent It W111 be shown sunultaneously that the equ111br1um d1str1but1on of an mterdeparture 1nterva1 1s exponent1a1, and hence 1t follows that the output d1str1but10n, or d1str1butlon of call completlons, IS P01sson PROOF IT SHOULD BE noted first that the probab111ty of the system bemg 111 some state k 1mmed1ately after a call departs 1s the same as the probablhty that an arrlvmg call Wlll find the system 1n state k A departmg call that leaves the system 1n state k represents a trans1t1on from state k+1 to state k, wh11e an arr1v1ng call’s findmg the system 1n state 19 1s a trans1t1on from state I: to state k+1 The number of trans1t1ons from state [C to state k+1 cannot dlfier by more than one from those from state k+l to state 10 1n any t1me 1nterval Hence the proport1on of calls leavmg the system 1n state k approaches the same hm1t as the proportlon of calls findmg the system 1n state 10 The latter hm1t 1s known to be equal to the equ1- 11br1um probab111ty of the system bemg 1n state k at an arbltrary mstant These equ111br1um probab111t1es comprlse the d1str1but1on,[4l Wh1ch 1n the present notatlon may be wr1tten Pk=Po (VIM/’9', (0§k<s) Pk=Po (MM/(sI 8""), (has) Where 170 IS determmed by the reqmrement 11:: Pk = 1 Let L denote the length of an arb1trary mterdeparture 1nterval and 702 Paul J Burke n(t) the state at a tune t after the last prevrous departure Let Fk(t) be the probablhty that n(t)=k and 101ntly that L>t , It may be helpful to note that 2:: mt) =F<z> 1s the fallmg dlstnbutlon of the length of an mterdeparture mterval and Fk(0)=Pk at equlhbrlum For an 1nfin1tes1mal 1nterval of length dt, Fo(t+dt) =Fo(t) (1 —>\ dt), w1th1n mfimtesunals of hxgher order, smce L>t+dt If and only If L>t and no calls arnve durlng dt Sumlarly, Fk(t+dt)=Fk(t) (1—x dt—Jp. dt)+Ft_1(t) A dt, where j=k for k<s and J=8 for has These equatlons reduce to the dxfferentlal equations Fn'(t) = “A 170(1), (1) [PI/(t) =)\ Fk.1(t) '— Owl-#1) Fla), subject to the 1mt1al condltlons (whlch Imply the emstence of equlhbmum) F140) =17); Equatxons (1) can be solved by Inductlon to yleld F10) =1»: 6—“ (2) as the umque solutrons subject to the 1mt1a1 cond1t10ns The result 1m- plles that the margmal dlstnbutxon of the mterdeparture intervals 18 ex- ponentlal W1th parameter A, re, the same as mterarnval d1str1but10n Also, the 1ndependence of L and n(L) can be readlly estabhshed from It The probablhty that t+dt>L>t and n(L+O) = It IS Fk+1(t) (k+1) u dt for k+1§s and Fk+1(t) s p dt for k+1 >s But these expressrons reduce to (woo (WY pa e‘“ A dt and [1/(31 8H» (x/p)‘ g»“ e‘” A dt respectlvely, whlch are factored mto the margmal probablllty functlons of n(L) and L, thus provmg the 1ndependence of L and n(L). The 1nde— Output of a Queuing System 703 pendence of the length of an arbltrary mterval and all subsequent mter— vals follows from the last result together wrth the Markoff property, as 1s shown by the formal argument followmg Let A represent the set of lengths of an arbrtrary number of 1nterde- parture mtervals subsequent to the arbltrary mterval of length L, and let P(——) represent the probablhty functron of the chance vanable(s) repre- sented mthm the brackets The Markolf property 1mphes P(Aln(L))=P(Aln(L), L) (3) where to av01d ambrgmty n(L) may be taken to mean n(L+0) The 1ndependence of ML) and L IS eqmvalent to P (n(L),L) = P(n(L)) P (L) (4) The Jomt probabrhty functlon of the m1t1al mterval—length, the state at the end of the mterval, and the set of subsequent Interval-lengths may be wrltten as P(L,n(L),A) =P(AIL,n(L)) P(L,n(L)) (5) Subst1tutmg (3) and (4) mto (5), one has P(L,n(L),A)=P(A|n(L)) P(n(L)) P(L) (6) =P(A,n(L)) P (L) Whence from (6), P(L,A)= 233:5” P(A,n(L)) P(L) =P(L) PM) From thls result follows the mutual 1ndependence of all Intervals, wh1ch concludes the proof It may be remarked that for the ethbnum output to be P01sson Ulll- formly for all values of the parameter p, the Input (assumed to have a fixed average) must be Pmsson T1118 follows from the fact that the output dlstnbutron may be made to approxrmate the 1nput by allowmg p to be— come 1nfin1te AN EXAMPLE As AN ILLUSTRATION of the apphcatlon of the result of 131115 paper, an Ideahzatlon of the Sltuat1on 1nvolvmg sales clerks and cashlers mentloned above Wlll be cons1dered It w111 be assumed that customers have access to any of the clerks who may be free and that the clerks have equal access to all the merchandrse After servwe by a clerk, a customer proceeds to cashlers and has access to all of them regardless of the type of purchase Also, there 1s a smgle queue In front of the clerks and another smgle queue 1n front of the cashiers SerVICe by the clerks and by the casluers 1s order- of -arr1va1 704 Paul J Burke It W111 be further assumed that ev1dence ex1sts that the serv1ce t1mes of clerks may be satlsfactorlly apprommated by an exponent1al d1str1but1on w1th average 1 5 mmutes, whlle the serv1ce t1mes of cashlers are almost constant at 1 mmute The problem 1s to determlne the numbers of clerks and cashlers necessary so that the probab111t1es of a customer’s belng de- layed more than three mmutes In front of the clerks or more than two mlnutes 1n front of the cash1ers are each less than 0 05 for a per1od of sev- eral hours durmg whlch customer arrlvals are random (Po1sson) W1th an average of two per m1nute Smce the serv1ce tunes of the clerks are exponent1al, 1t may be 1nferred from the theorem of th1s paper that the 1nput to the cash1ers 1s P01sson w1th an average of two per m1nute regardless of the number of clerks, pro— Vlded only that these exceed three, and hence the number of cash1ers can be determmed 1ndependently of the number of clerks The requ1red number of clerks can be found W1th the a1d of plotted delay d1str1but1ons for exponent1al hold1ng tlmes [51 For five clerks, the occu- pancy 1s 0 60, s1nce arnvals average three per serV1ce t1me, and the prob— ab111ty of delay greater than two serv1ce t1mes (three mlnutes) 1s found graphwally to be 0 0047, wh1ch more than meets the cr1ter10n For four clerks, however, the occupancy 1s 0 75 and the probablhty of delay greater than two serv1ce t1mes 1s 0 070, whlch does not meet the crlterlon Hence five clerks are necessary Smnlarly, to determme the number of cashlers, plotted delay (hstnbu— tlons for constant hold1ng t1mes may be used 1‘” If there are three cashlers, the occupancy w111 be 0 67, and the probablhty of delay greater than two serv1ce tlmes W111 be less than 0 01 Hence five clerks and three cash1ers W111 be requ1red to meet the serwce cr1ter1a ACKNOWLEDGMENTS THANKS are due to W S HAYWARD, JR for suggestmg the method of d1fferent1al equatlons, to E REICH for cr1t1c1sm and correctlon of part of the proof, and to R H DAVIS for helpful d1scuss1ons REFERENCES 1 PHILIP M MORSE, “Stochastlc Propert1es of Wa1t1ng Lmes,” Opns Res 3, 256 (1955) 2 GEORGE G O’BRIEN, “Some Queumg Problems,” J Soc Indust Appl Math 2, 134 (1954) 3 R R P JACKSON, “Queuemg Systems w1th Phase Type Serv1ce,” Opnl Res Quart 5, 109-120 (1954) 4 T C FRY, Probabzhty and Its Engmeenng Uses, 382, Van Nostrand, New York, 1928 5 E C MOLINA, “Apphcatlon of the Theory of Probab1hty to Telephone Trunk- mg Problems,” Bell System Tech J 6, 461—494 (1927) 6 C D CROMMELIN, “Delay Probab111ty Formulae,” Post Ofiee Elec Eng J 26.. 266—274 (1934) Copyright 1956, by INFORMS, all rights resenred. |Copyright of |CJIperations Research is the property of INFORMS: Institute for |CJIperations Research and its content may not be copied or emailed to multiple sites or posted to a listsenr without the copyright holder's express written permission. 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The output of a Queuing System - THE OUTPUT OF A QUEUING...

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