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Priority Queuing Networks By RJ. T. MORRIS (teerusctrecoved October 17, 1860) Priority service disciplines are widely used in computer and com: ‘munications systems, Many such systoms can be modeled by quewing networks, bul preseny developed Ur does not allo solution of ‘there models when priority sericedistiplinen are present. Por priority ‘queuing networks that have a homogeneity propery, we give some ‘explicit results for mean delay and throughput. However, the as ‘numption of homogeneity is too restrictive for many applications, We identify some examples of systems or whieh inhomogeneous teo-node ‘Priority queuing networks are appropriate models a yield to exact ‘analysie The esulteallew some conelusiona tobe drain abort using ‘rioritee in a teo-node closed network to extablish grader of service. We also use the results to evaluate a commnonty used appresimation ‘technique for priority queuing systems, |. INTRODUCTION AND SUMMARY Priority service disciplines are widely used in computer and com munication system One common application af priorities ie in the establishment of multiplo gradee of service whereby deferable or background work is scheduled according toa lower priority. In other applications, a device may give prioritized service to clas of jobs Keown to be shart so aso Inerense nverall system throughput: Fer rpurpoves of performance analysis, commuter and communication tee tears have often been modeled as queuing networks. However, the theory of queuing nevworks ines present form (se lef 1,2} doesnot provide solutions for even simple noowoess with prionty disciplines. fexcopt in an approximate manner! * "There are very fi exact rel far peuing networes Gis, queuing snurlely with more than one service staion ar nate with priiy to be found in the erature, One known reeule concerns a general service time, ingle-server queue with preomprive or nanpreemptive priority fd Fnice exponential source! This model can he tought of a8 a 90 node closed queuing networle where the eecond node isa pure delay or infiniteserver group. Ina paper by AviTezhak and Hoyman the mean ‘yee times were abesined fur w central server model with priorities, {vr the assumption Una the mean service times nd routing pattems tre the sane foreach priority laa" In othor computer and commu fiction applications, nlwork priorities have onl’ been represented fpprowimately. using heuristics forthe central server model and a petkelavicching network” While these approximation techniques may be adequate in scoriey forthe parameter ranges of some applientions, ‘much further work is needad in improving and validating these tech rigues and ultimely, i developing exaet analytical results whensver they evo tractable ‘Our gots sre to obtain insight into the aolucion form for some simple ceases of quouing networks with prioriti,co obtain an inital evaation of the accuracy of existing approximation techniques, and to drsw Some conclusions on the performance of some simple network priority Structures oocurring in prastice. In Section TE, we describe a general Class of priority quewing networks (hat ate homogeneous in the sense thet all customer classes are treated identically with rospect to service timo and routing. For homogencous networks, we are able to give tneen delay and througlpue analysis, However, it wll be sean tht the hhamogeneity asumpton i auficenty resirietive as to prevent application of these rotlts in many situations. bsequently, we focus on ‘pro specife examples of spatems Ut can be modeled by simple “Queuing networks, but in which priority disciplines and inhornogeneity play crucel rolex These examples suggest several different two-noWde priority quoting network models that eld vo exact analy "The first example we consider sa computer system consisting af a contra processing unit (ere) and an input/output (1/0) deve, which Droceases both fime-riial transaetions, at well aa nontime-critical batch jos ‘The system ie designed wo glve priority to the transactions At both the cPU and 170 deviee (in contrast Refi. 8 and 4 where it in anmumed that only the c#t observes priorities). This suggests use of the twornode, closed queuing network model A of Fig. 1, wich one ‘ode representing the CFU, and the other. the 1/0 device. The modal thas separate quetes foreach privity class at each node, and procty is obeerved preemplively at boch nodes. ‘The second example » fal-duplex data Unk whichis usod for tranemission of messages under # indow flow control protocol There ttre bwo grades of metsages, the premium grade and the standard fade, When hoch premium yade meceages and acknowledgments Feceive preemptive priority, model A is applicable (refer to Fig. ‘which is explained further in Seclion VI), However, since acknowled- Tents exe typically shorter than messages, another conigurion it 1748 THE BELL SYSTEM TECHNICAL JOURNAL, OCTOBER 186: @ — ae) Be) car oo “ suggeated wherein acknowledgments of either grade are given preemptive priority this lads to model B shown in Fig 1h Tn both models A and B there are u fixed mumber of customers in ‘each clase and service time diserhutions at » nade nresuaimed to be ‘exponential, but are not required toe the sane a moc for wach ‘curtomer class (in contrast with homogencoua networks or che fir ‘omefirst-erved nodes describe in ue), Bees ofthe exponential ‘asitption, the priorities can be understood tobe ether preniptive- ecume of precmptive-restar (with exarpling) Tor each model, et ‘ice within a customer class at node ran he thought of as first-come first-served, but all equations and reals remain valid for any other dtacipline within the priority class which dows not take into account the actual sorvie time requirement when selecting » customer for "The general approach we use in the analysia of model A an Rand several similar mes, sto st up the balance equations (steedy-state ‘Kolmogorov forward equations) for the Markov chain describing the ‘numberof customers of each privity clas: at ene ned, These pata difference equations generally do Hot satisfy thr well known. local hulanee condition ” but, nevertheless, can sill be solved to obtain the stationary distribution, This disrbaion allows throughput awl mewn delay vo be computed for each customer class. ‘These results can he applied to ubtsin some general conclusions bout the bro system we have uscd ws motivating examplos, Tn the ‘ompucer ayetem that processes both transetions and batch jobs, we Fin that if the transactions are botteneckedl atone device (CPU or 1/ Ol the batch jobs new! to be aven moro strongly bottlnecked atthe thor device, if @ significant belsh tbrougbput is to be attained. Spociely, we ahow that if che cranzectons bave w boleneck of eength sat one node, the batch jobe need to have a hottleneek of rength ¢" at the other node (where fe the traraaction mukipre framing level), ifthe batch jobs are to be able to fulfil a role at ville" work. In the dala lnk example, we find 2 similar result: If standard grade message traffic ie carried in purely background mode, Thinly extreme parameters are necesazy before ita intoduetion be: comes attractive, On the other hand, some compromise of the Dromium rfc performance ie perited, chen an appreciable anount Of wandard grade meseaye rfc ean bo carsiod by using daca link ‘Capacity that would otherwie be wasted, Fr each system, we identify hazards thet can accur when the Towerpririty work is allowed to interfere with higher-pricrty work. Refer to Sections V and VT far farther detail In Section VIL, we use the results to evaluate tho effectiveness of « well Anon approximation technique. We find that acruraey of the “pproximation technique varies fom good co poor. depending on the Darameters of the application. 8 cxterion on The application param: fers ia proposed under which the approximation ceehnigue would be fexpected to pertorn wall Il. HOMOGENEOUS NETWORKS In cis section, we consider elas of queuing avtworks that allow preemplive priorities but a otherige homoguneous in che sense that tt any one node al customers nev teeated identielly with respmet to Service rate and routing The neulta rely on an observation similar to {hat made by Wiltshak and Heyman in thelr analysis of he central server model” ‘We fs. conser a clad quouing network of the Gordon-Newell typos It consists of eervi eos oF nades numbered 1, «+», N.Tn departure (rom the Gordon-Newell formulation, there are P priority lasses numbered f ==, 2 with the ith (priors) clas containing K, customers, (= 1, +, Pant UReK ‘At any node, w cutoinee from a higher wxelered class takes preomp- five prionily overs custouner hom n lower numbered elass."Tho sarvie 1749 THE BELL SYSTEM TECHNICAL JOURNAL, OCTOBER 1281 time distribution st node ie exponential with rate forall eastomers, snd the service discipline i fir-cene First served within exch priority clase Aer service a @ node is completed, routing to another node ix governed by a probability veccr which isthe same for each priority la. Let the state of the network be expressed by the quantities Fat cee Noam ty von, P, where n; denotes the number of lass “astomers present af node, Define the aggeewale sate variable min 3 ni, the number of priory els {ur higher stoners at node j The key ‘observation is ht the random variable ms cquiveent vo that which ‘would result f the network were modified by first removing customers fot priority Te than 1 (Le, hy aelting Rem 0,1 = kee) and thereaer, ignoring all priority terior distinctions. ‘The is because () lower Priority customers exer no influence on highee-piortyeustomers, nd {G9 regurdle of whether priorities aresibserved between customers of assent, Ih =-+ P latins in the otal wimber (mn) of estomers Df priority j, oF larger, at a node are not altered (by the assumed ‘ulnemity of service rate and wouting over priority eases), This, we "an find the stationary discibution of ou; hy the us lowe qung ‘network techniques,“ The stationary distribution of the aggregate ‘arial nici to determine the senlyatate mean delay and Uewuglpal of each priority clase al each nae Thi follows from the fact that foreach én Fn] = Fm!) ~ Bim = 0] = Prtm} >t — Print > 0), Pen} > 0, where me!" ie understuod tobe identically aero. Hence, priority clas Peastomers have a throughput st node j of 1) abel 0) ~ Primi" > 09] and # mean delay (including servie time) of B= (Bim) ~ Bim VT, by Litee’s Law. Note that chesa quantities are obtained in the procs of carrying out the mean value analyis for a single-chain closed fetoonk with K customer, ‘Similar relts are obtained for an open necwork of the Jackson, type All notation i the same at for Une lowed network, except tha swe na Tanger specify the number of custonners of each priority clase Dut, instead, we speify the rate A} of exogenous Poisson arrivals of privity else 2 to node J. We must now asrume chet the trafic QUEUING NETWORKS 1749 ‘equations admit « unique solution e! representing the mean anival tority ela to node and that ddsw for each j, We make the wosloyous observation that the quantity mj can be oblained by considering the network modified by turning uff favival stream of priority leat than é (ce, aetting XP = 0, = B <j, 1N) and, hereafter, ignoring priority discinctions at service. Wet thn hve Blo] = Foot] ~ Bb)” Teh and DS = (Bim) ~ Fim"Yef Now, and, therefore, where p= ef/) i the utilization of node j cue 10 class & customers ‘We, thas, recognize the validity in a neework context of the Cobham: ‘ype formula orginally obtained by White and Christi for tn: dlay inven Wolated preempive priority M/M/1 queue when the service Limes are che sume for exch prinity clas "Within the stringent limitations imposed by our homogeneity assumptions, some externa to these results are posible, For example, Wwe ean allow the more general servico disciplines shown by Kelly (0 Fel. 2, pages 6# and 78) to lead vo preduct form provided (0) the state Alependence and server sharing embodied in these disciplines extend ‘only to the customers of higher priority present ata node and ignore fl lower priority customers, and (i) all customers are created ident tally with rempect to eervce time and routing ‘The abovo results mig pesiby bo wef ia some queuing network applications, For example, Gre-cut evaluation of the impact of introducing data packel. pierces ito w packet switching network ‘ould be carried out by representing exogunous packet arrivals ax PPissom and data links ss expementia servers" and by assuming that the mean data packet leaulh snd the traffic routing paliern are the 1750 THE BELL SYSTEM TECHNICAL JOURNAL, OCTOBER 1881 ‘ame for each priority late If short contzol (eg, aeknowiedgment) packets were tobe given prcrty over data packets, we find chat our hhomogensiy assumptions would be violate although the effect of 0 shore packets could be approximated in several ways! Indeed, the ‘ase of control packets receiving priority serves to iliostrate a common fituation in which customers receiving priority have sgniicantly ‘mallr sevice time requirements Thus, the results described inthis section are expected (o find limited use. The remainder ofthis paper does not make any such homogeneity assumption: unfortunstaly, by relaxing the wsampion, we are uble Wo reat only networks consisting of two nodes, Mi, MODEL A: TWO NODES WITH PRIORITIES THE SAME AT EACH NODE. ‘We now consider the two node closed qucuing network introduced {in Section I as model A and shown in Fig, 1a. The network consists of tio nodt—the left. and ight hand nodes. There ore N high-priority and Mf low-priarcy customers. High-priovity customers take preem- ‘ive priority over low-prioity customers ac ench node, Allserice times are tasumed exponencally distibuted: the high-priority customer Ihave a mean service ime of + a th left node and Tat the right low priority customers have a mean service lime ofv "at the left node and Av ac the right, After service at one node ia completed, cuetomers are immediately routed co the other mule without changing lass, We anaume 2, Nand AF are all pastve. The State af the system in desribed by the vector fu) where (respectively mm is che number of high- (eapectvely low) priority customers t the left nae. "The sco (n ml evlves asa Markoy chain ‘with stationary distribution pin, Tis obvious thatthe stationary Gisrbution is alto The Filing distbution since the chain ie nite land inedacile. The teansitions of (3m) sw suns in Fig. 2 Hy definition, inom atten Ue tr en Pla, mO[Linay + Hl any Hoes + yaa = pin =, m) + pin + mie plN, m= TIM any +pl0,m+ thle, OFREN, O=m=M, (I here |, , denotes the indicator function which has value I (respec: tively 0} when the predicate within the braces is true (respectively fale). Note that wo are adopting the convention that pin,m) = O when n,m) €[0,N] x [0 MY ‘We wish (ool for i The techniaue we se is best explained by reference 10 the state transition diagram shown in Fig. 28, FUst (QUEUING NETWORKS 1781 ate eson iaerey rel sm ee = 4H ~ 30) State ARETE Uta noopecptives thors or = nove that for any 7 = 0, pln, m), <= is expressible in terms of pin in), O2m=M. ence, pl, m),0=2m 2M, 0< n= N ean be Cxpresed in terms ofthe To boundary values p10, =m = Mf and falution for p00, m), 0 m= 7 rests on the balanee eations for the ht boundary ptm), =m ==, Ths observation is generally crue for sn sbitrery two-dimensional birth-death prowess (provided all ‘ight to-left horiwntal transicons are preset] but not always usoful fine the reaillane squstions for p(0, m), 0 = m = Mare no easily tlved. Fortunately, im our eave, the absence of vertical transitions fo states (4, m),0-< m= N eatned by the priority strucure revults {nrelations fr 710, m) which comprise & simple diffrence equation of fonder ovo which i aay aolved and yilds an explicit solution for (n, in). Before proceeding with ths (echnique, we mention that a compu {ational method based on such an observation hs hoen proposed in ‘which the recursive siructute a used to reduoe the problem of finding the stationary distribution of certain N 2M birtheath processes the rlution of N equations nN unknown For nocational ease, define aim) = (0, m), = m <= A. Waiting (I) for n=O yield ALO) = (0)! eli @ Alm) = atmiu+ te alm + tw" O<m<M DAL AO = aN + Te o and for = n= pn mir -+2) pln + Lyme + pln t,m), Osm=st 6) 1752 THE BELL SYSTEM TECHNICAL JOURNAL, OCTOBER 1961 for which the general solution is nad = fame" = 9°") + p,m. — oan 6 provided » # 1, Henee, the problem is reduced to determination of ‘lm, 0m = Mf. Thi is done by writing eq (1) for n=, DAN, ON +) = PIN ~ 1,0), 0 PAN, mye) = PV Lem) +pIN.m WR, Oma, (8) BAN, My = pl — 1, MY + ol, MR o Substituting en. (2) and (6) into on. (7, (1/00) = fale Me Yee A= 1 =A"), (10) Assuming M > 1, taking mt = 1 in og. (8, and using ege. (2), (8), and (6) yields a(2)fat1) =» where prwry tee frst a Note that the denominator of + ix not zero Beeause » #1. Taking Tem finer 8) will gs. (and (6) yea the difference equation cm + fake 4 ah wd + nue Ba Day Be” — As — Fel = OPW + AES — Ae ae] =O, Lm hag characteristic woot 1,7 But since wi2}/ullh = 7, we have mast. or) 1t can now he verifed that ¢hncr roclts nee cnnastent withthe one ‘unused oq, (9) and thatthe reault holds tue for Mf = 2 ‘Subeliiting eae (2 to 4 (10), and (12 int og () and simplifying {cds the general solution for» 1 atm} = att 1 Umm) = Crfre sot — I g(t S1-ne", OsneN, Osms¥, (13) where ris given by eq. (11) and 8) i the Kronecker delta: 6() = SUA) = 0, 40. The normalizing constant Ci ublained by demanding that pin, m), O= n= N, 0 m= af isa probability distribution, yielding awe) a-eeyfa a Ze ew —o) as QUEUING NETWORKS 1753 In en. (14) and below we refrain from summing geometric series of the fore Be to avo m special tatement for the case ‘Our treatment las excluded the ease » = 1 for which the alution to eq (6) 8 pln, = adn (1— n) + pC, mn. Rather chan resolving for this casei i ensir to eae the Limit 1 in a (18) and (14); thi Inst yield the correet solution since the coefficients in eq. (1) are continuous inp and the eigenvectors of w malrix are continuous functions of ils elements. Hquations (13) and (14) for pin, m) can be rewritten ina form which i welldefined for #= plasm) = De") 4 igs whore a8) i and summations over descending ranges are taken wo be 2e70, Since the solution given hy eqs. (15) to C17) is continuous in »> 0, is the seneral solution for all> 0. (Tho system considered here can be generalized trivially to allow mesn service vie of the high-priority customers atthe right ‘node to be «(rather than unity} and eo allow routing of « eustomer beck to the node nt which it ha just completed, Let high-priority customers.on completion atthe left iespestvely right) node be rosted to the right (renpectvely lft) node with probability (Respectively ie) and be routed to thy torent which completion has just ovcurred ‘with complementary probity 1~ pe (eapeetivly 1 ~ p) Lot the low-priorty customers have similarly defined probabilities diy gu ‘Then the results ines, (23) lo (17) renin valid with », a) replaced, respectively, by xpos strlen Mads (Gi) Tesezondlly verified that the marginal distribution of tho high priority eustamers pin, -) =3 » ln, m) agrees with the result for an ordinary twornode coved network, vi, 4784 THE BELL SYIEM TECHNICAL JOURNAL, OCTOBER 1961