Research Article Control of Traffic Intensity in Hyperexponential and Mixed Erlang Queueing Systems with a Method Based on SPRT

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1 Mathematical Problems i Egieerig Volume 23, Article ID 2424, 9 pages Research Article Cotrol of Traffic Itesity i Hyperexpoetial ad Mixed Erlag Queueig Systems with a Method Based o SPRT Müjga Zobu ad Vedat SaLlam 2 Departmet of Statistics, Amasya Uiversity, 5 Amasya, Turkey 2 Departmet of Statistics, Odokuz Mayıs Uiversity, 5539 Samsu, Turkey Correspodece should be addressed to Müjga Zobu; mujgazobu@hotmail.com Received 6 December 22; Revised 3 May 23; Accepted Jue 23 Academic Editor: Wuqua Li Copyright 23 M. Zobu ad V. Sağlam. This is a ope access article distributed uder the Creative Commos Attributio Licese, which permits urestricted use, distributio, ad reproductio i ay medium, provided the origial work is properly cited. The cotrol of traffic itesity ρ) is oe of the importat problems i the study of queueig systems. Rao et al. 984) developed a method to detect chages i the traffic itesity i queueig systems of the M/G/ ad GI/M/c types based o the Sequetial Probability Ratio Test SPRT). I this paper, SPRT is theoretically ivestigated for two differet phase-type queueig systems which cosist of hyperexpoetial ad mixed Erlag. Also, for testig H :ρ=ρ agaist H :ρ=ρ, Operatig Characteristic OC) ad Average Sample Number ASN) fuctios are obtaied with umerical methods usig multipoit derivative equatios accordig to differet situatios of α ad β type errors. Afterward, umerical illustratios for each model are provided with Matlab programmig.. Itroductio There are very few studies o the sequetial detectio of parameter chages of queueig systems i the literature. The theory of SPRT for a sequece of observatios formig a fiite Markov chai was give i []. After, based upo the theory of [] was discussed statistical quality cotrol ad SPRT procedures for the cotrol of traffic itesity i [2 6]. This method aims to detect chages i traffic itesity by observig oly the umber of customers i the system at successive departure epochs Q,whichareembeddedMarkov poits. Recetly, a SPRT to regulate the traffic itesity based o the umber of arrivals durig the th service periods for M/E k / queue was proposed i [7] adaautoregressive process based o the umber of customers at the departure poit ad its applicatio to the queueig model were give i [8]. I this paper, the method based o the SPRT is first examied theoretically for the two differet queueig phase types. Afterward, the observatios obtaied for differet samples model are evaluated for each of the models by simulatio through MATLAB 7.. R2a) programmig. 2. Queueig Process ad SPRT I may queueig applicatios, a performace characteristic of great importace ad iterest is the traffic itesity ρ the ratio of mea arrival rate λ to mea service rate ). The purpose of this test is to determie as quickly as possible the chages i traffic itesity ad to take the appropriate corrective actios. With this objective, a procedure has bee developed for testig the hypothesis H : ρ = ρ agaist H : ρ = ρ usigwald ssprtforthesystemsm/g/ ad GI/M/c, i which queue legth processes have imbedded Markov chais {Q } i [6]. This procedure is explaied as follows. Cosider the sigle server queue where arrivals occur accordig to a Poisso process with rate λ per uit time. The service times of customers are idepedet idetically distributed i.i.d.) radom variables with the distributio Bx). For this system, the queue legths at service completio poits form a imbedded Markov chai. Q will be the umber of customers left behid by the th departig customer. The capacity of the queueig system is restricted

2 2 Mathematical Problems i Egieerig to N. The the trasitio probability matrix of the imbedded Markov chai is give by P={P ij ρ)} = 2. N 2 N k k. [ [ k k k. k 2 k 2 k k. k 2 k j k 2 k j k 3 k j k, ] ] where k = P { arrivals durig a service period} = e λt λt) j /j!)dbt). Let t <t <t 2 <<t be the set of time poits at which the Qt) process exhibits the Markov property; amely, P{Q t + )=j\q t )=i,q t )=i,...,q t )=i } =P{Q t + )=j\q t )=i}. I the M/G/ queue, t is the time of service completio of the th customer ad, i the case of the GI/M/c queue, t is the arrival epoch of the th customer. I the case of the M/M/c/N queue, though the process is i Markovia cotiuous time, for purposes of cotrol it is easier to examie the process at arrival poits. For simplicity, we will use the otatio Q = Qt ). Deoted by P = {P ij ρ)} the trasitio probability matrix of the Markov chai {Q } defied over state space E = {,,2,...,N},where ) 2) P ij ρ) =P{Q + =j\q =i;ρ}. 3) It is assumed that P is kow except for the value of the parameter ρ. The problem is to test the hypothesis H :ρ=ρ agaist H :ρ=ρ. Cosider the sequece of observatios Q, Q, Q 2,...,Q. The joit probability of observig this sequece uder H ad H is give by Pr {Q, Q, Q 2,...,Q a ;ρ i }=PQ ;ρ i ) j= PQ j \ Q j ;ρ i ), i =,. 4) The the likelihood ratio is L= PQ ;ρ ) j= PQ j \ Q j ;ρ ) PQ ;ρ ) j= PQ j \ Q j ;ρ ). 5) Z = l PQ ;ρ ) PQ ;ρ ), 6) Z r = l PQ r \ Q r ;ρ ), r ). 7) PQ r \ Q r ;ρ ) Let A = β)/α ad B = β/ α),whereα ad β are theprobabilitiesoftheerrorsofthefirstadsecodtypes. The, Wald s SPRT [9]totestH :ρ=ρ agaist H :ρ=ρ becomes as follows. Observe {Q i } i =,, 2,...) successively ad at stage, ) accept H if Z r l B; 2) accept H if Z r l A; 3) cotiue by observig Q + if l B< Z r < l A. If we assume Q =i is specified ad deote by ij the umber of trasitios i j up to ad icludig the th trasitio, the the likelihood ratio give i 5) reducesto L= P ij ij ρ ) P ij ij ρ ), l L= ij l P ij ρ ) P ij ρ ). ThetheSPRTfortestigthehypothesisH : ρ = ρ agaist H :ρ=ρ will have its cotiuatio regio l B< ij l P ij ρ ) < l A. 9) P ij ρ ) I the case of M/E k /, E k /M/c, adm/m/c/n queues, we will show i the sequel that the logarithm of the likelihood ratiocabewritteitheform l L=a+ ij c ij, ) where a, c ij, ad P ij are costats depedig upo the parameters ρ, ρ, ad the trasitio probabilities P ij.thus 9)reducesto l B a< ij c ij < l A a. ) From this it follows that the cotiuatio regio of the test is bouded by straight lies l A ave l B a. 3. Operatig Characteristic ad Average Sample Number Approximate formulas for the OC ad ASN fuctios are givei[]. I order to evaluate the OC ad ASN fuctios, 8)

3 Mathematical Problems i Egieerig 3 λ t, ρ) is determied as the largest real positive latet root of the matrix P t) = {p ij ρ)[ p t ij ρ ) p ij ρ ) ] }, 2) its derivate at t=ad the ozero root tρ) of the equatio λ t, ρ) =. A umerical search techique was employed to fid the root i []. For evaluatig the derivate of λ t, ρ) at t =, the followig five-poit umerical differetiatio formula has bee used: λ t, ρ) 2h {λ 2 8λ +8λ λ 2 }, 3) λ t, ρ) 2h 2 { λ 2 + 6λ 3λ + 6h λ 2 }. 4) Whe the stated space of {Q } is fiite, the OC fuctio forthesprtcabeobtaiedas Lρ) A t ρ) A t ρ) B t ρ) if t ρ) = l A l A l B if t ρ) =, 5) where t ρ) is the ozero real root of the equatio λ t, ρ) =.TheASNcathebeobtaiedas E ; ρ) Lρ)l B + { L ρ)} l A λ ) if λ = 6) Lρ){l B}2 +{ Lρ)}{l A} 2 λ 4. SPRT for Phase-Type Distributio if λ =. 7) The expoetial distributio is very widely used i performace modellig. The reaso, of course, is that mathematical tractability flows from the memoryless property of this distributio. But sometimes mathematical tractability is ot sufficiet to overcome the eed for a model process i which the expoetial distributio is simply ot adequate. This leads us to explore ways i which we ca develop more geeral distributios while maitaiig some of the tractability of the expoetial. This is precisely what phase-type distributios permit us to do []. 4.. Hyperexpoetial-2 Service Model: The M/H 2 / Queue. Cosider the cofiguratio preseted i Figure, iwhich α is the probability that the upper phase is take ad α 2 = α the probability that the lower phase is take. If such adistributioisusedtomodelaservicefacility,thea customer eterig a service will, with probability α,receive the service that is expoetially distributed with parameter ad the exit the server or else with probability α 2 receive the service that is expoetially distributed with parameter 2 ad the exit the server. Oce agai, oly oe customer α α 2 Figure : Two expoetial phases i parallel. cabeitheprocessofreceivigserviceatayoetime;that is, both phases caot be active at the same time []. Theorem. The desity fuctio of the service time is give by db x) dx I this case, oe has 2 =α e x +α 2 2 e 2x ), x ). 8) k =α ρ ρ + ) ρ + ) +α 2 ρ 2 ρ 2 + ) ρ 2 + ), where k = P { arrivals durig a serviceperiod}. Proof. Cosider e λt λt) k = α! e t +α 2 2 e 2t )dt) = + = α λ! e λt λt) α! e t d t) e λt λt) α! 2 2 e 2t d t) + α 2 2 λ! = α λ! + α 2 2 λ! e λt t e t d t) e λt t e 2t d t) e tλ+) t d t) e tλ+2) t d t) = α λ + ) Γ +)! λ+ + α 2 2 λ + ) Γ +)! λ+ 2 9)

4 4 Mathematical Problems i Egieerig Here = α λ! + )! λ+ + α 2 2 λ! + )! λ+ 2 =α λ ) )+α λ+ λ+ 2 λ ) λ+ 2 λ/ =α ) / ) λ/ + / λ/ + / λ/ +α 2 2 ) 2 / 2 ) λ/ / 2 λ/ / 2 2 λ+ 2 ) =α ρ ρ + ) ρ + )+α 2 ρ 2 ρ 2 + ) ρ 2 + ). P ij ρ) = k j i+, j=,,...,n, P j ρ) = k j, i=,2,...,n, j i, j=,,...,n, N P N ρ) = N i P in ρ) = N i P in ρ) = r= k, k, i=,2,...,n, α ρ ρ + ) r ρ + ) +α 2 ρ 2 ρ 2 + ) r ρ 2 + ). 2) 2) With these values for P ij ρ), the logarithm of the likelihood ratio will be l L= ij l P ij ρ ) P ij ρ ), l L= ij l α ρ ) ρ ) + ) +α 2 ρ 2) ρ 2) + ) α ρ ) ρ ) + ) +α 2 ρ 2) ρ 2) + ) ρ ) + ) ρ 2) + )) ρ ) + ) ρ 2) + )) ), 22) where c in c ij = l α ρ ) ρ ) + ) j i+ +α 2 ρ 2) ρ 2) + ) j i+ α ρ ) ρ ) + ) j i+ +α 2 ρ 2) ρ 2) + ) j i+ ρ ) + ) ρ 2) + )) ρ ) + ) ρ 2) + )) ), i=,2,...,n, j=,,...,n, j i, c j = l α ρ ) ρ ) + ) j +α 2 ρ 2) ρ 2) + ) j α ρ ) ρ ) + ) j +α 2 ρ 2) ρ 2) + ) j ρ ) + ) ρ 2) + )) ρ ) + ) ρ 2) + )) ), j=,,...,n, = l { N i { α ρ r ) r= ρ ) + ) ρ ) + ) { N i r= +α 2 ρ 2) ρ 2) + ) r α ρ ) ρ ) + ) r +α 2 ρ ) ρ ) + ) r c N =c N. ρ 2) + ))) ρ ) + ) ) } ρ ) + )) }, } i=,2,...,n, 23) 4.2. Mixed Erlag-k Service Model: The M/mixE k / Queue. ThisisillustratediFigure 2, where,withprobabilityα, the top series of k expoetial phases is take ad, with probability α, the bottom series of k phases is take.

5 Mathematical Problems i Egieerig 5 α 2 k + αλ k) k! k )! e tλ+k) t +k d t) α 2 k k Figure 2: Mixed Erlag represetatio. = kλ k k )! k )! e tλ+k) t +k d t) Gama distributio Γ +k) Theorem 2. The desity fuctio of the service time is give by = αλ k) k! k 2)! λ+k ) +k Γ +k ) db x) dx =α x)k 2 e x k 2)! where k 2is a iteger. I this case, oe has + α) x)k e x, x ), k )! k = ρ ρ+k ) k k ρ+k ) +k 2)!! k 2)! α+ α) +k ) k ) k ρ+k )), 24) 25) + α) λ k) k +k! k )! λ+k ) Γ +k) = +k 2)!αλ k) k +k! k 2)! λ+k ) + +k )! α) λ k) k +k! k )! λ+k ) = +k 2 ) λ+k ) k λ+k ) αk) k λ + +k ) λ+k ) k λ+k ) α)k) k λ where k = P { arrivals durig a serviceperiod} = e λt λt) j /j!)dbt). Proof. Cosider e λt λt) k =! α kt)k 2 k 2)! e ktx k+ α) kt) k k )! e kt k)d t) = e λt λt)! α kt)k 2 k 2)! e ktx k d t) e λt λt) + α) kt)k! k )! e ktx k d t) = αkλ k) k 2! k 2)! + α) kλ k) k! k )! = αkλ k k 2)! k 2)! + αkλ k k )! k )! = αλ k) k! k 2)! e λt t e kt t k 2 d t) e λt t e kt t k d t) e tλ+k) t +k 2 d t) e tλ+k) t +k d t) e tλ+k) t +k 2 d t) =α +k 2 ) λ λ+k ) k k λ+k ) + α) +k ) λ λ+k ) k k λ+k ). If the umerator ad deomiator are divided by, k =α +k 2 λ/ ) λ + k) / ) k) / λ + k) / ) k 26) + α) +k λ/ ) λ+ k) / ) k) / λ + k) / ) =α +k 2 ) ρ ρ+k ) k k ρ+k ) + α) +k ) ρ ρ+k ) k k ρ+k ) adcobracketsaretakeasfollows: k = ρ ρ+k ) k k ρ+k ) α +k 2)!! k 2)! + α) +k )+k 2)!! k )k 2)! k ρ+k )) k 27)

6 6 Mathematical Problems i Egieerig Here = ρ ρ+k ) k k ρ+k ) +k 2)!! k 2)! α+ α) +k ) k ) P ij ρ) = k j i+, j=,,...,n, P j ρ) = k j, N i P in ρ) = k ρ+k )). 28) i=,2,...,n, j i, j=,,...,n, N P N ρ) = k, k, i=,2,...,n, 29) where ρ ρ +k ) k ρ +k ) k k =k ) l ρ +k ρ +k ) α + α) +k ) k ) + ij l ρ ρ +k ρ ρ +k ) + ij k ρ +k )) ) ), 3) l α+ α)+k ) / k )) k/ ρ +k)) α+ α)+k ) / k )) k/ ρ +k)) ). 3) N i = r= ρ ρ+k ) r k k ρ+k ) α+ α) r+k ) k ) +k 2 ) k ρ+k )). Here a=k ) l ρ +k ρ +k ), With these values for P ij ρ), the logarithm of the likelihood ratio will be l L= ij l P ij ρ ) P ij ρ ) l L =a+ ij c ij, = ij l ρ ρ +k ) k k ρ +k ) +k 2)!! k 2)! l L α+ α) +k ) k ) k ρ +k ))) ρ ρ +k ) k k ρ +k ) +k 2)!! k 2)! α+ α) +k ) k ) = ij l ρ ρ +k ) k k ρ +k ) α+ α) +k ) k ) k ρ +k )) ) ), k ρ +k ))) c ij +b ij = l ρ ρ ρ +k ρ +k ) j i+ + l α+ α) j i + k) / k ))k/ρ +k)) α+ α) j i + k) / k ))k/ρ +k)) ), c j +b j = l ρ ρ j +k ρ ρ +k ) i=,2,...,n, j=,,...,n, j i, + l α+ α) j+k )/k ))k/ρ +k)) α+ α) j+k )/k ))k/ρ +k)) ), c in = l { N i ρ r ρ +k ) k k ρ +k ) r= α+ α) r+k ) k ) N i ρ r ρ +k ) r= k ρ +k ) k j=,,...,n, r+k 2 ) r k ρ +k )) ) r+k 2 ) r

7 Mathematical Problems i Egieerig 7 5. Numerical Results α+ α) r+k ) k ) c N =c N. k ρ +k )) ) } } } i=,2,...,n, 32) Cosider a M/H 2 / queue with poisso arrivals, two-phasetype expoetial service, fixed N). Let α =.4 ad α 2 =.6 be the probability for the upper phase ad the probability for the lower phase ad α =.5 ad β =.5 the first ad secod types of errors, respectively. Let N=4be the capacity of the queuig system. The mea value of ρ is calculated usig the followig formula: ρ= λ α / )+α 2 / 2 )). 33) Suppose that we wish to maitai ρ at the level.5 ad we wish to detect whether its value has icreased. The, the hypothesis test is H :ρ =.5agaist alterative H :ρ =.8. Let t =,t,t 2,t 3,...be a discrete set of the umber of customers remaiig at poits of departure i the M/H 2 / queue. The umber of customers remaiig at the 48-poit departure is give as follows: ) State space is E = {,,2,3}. Table shows ij that the umber of trasitios i j,adtable 2 shows the value of c ij calculatig for SPRT. From l B< ij c ij < l A decisio regio, ij c ij is betwee l A ad l B so that <.62 < Therefore, the system is observed agai ad the umber of customers remaiig at the 49 poits of departure is give as follows: ) Table 3 shows ij that the umber of trasitios i j. From l B < ij c ij < l A, the decisio regio, ij c ij,issmalllb so that < Therefore, the hypothesis H is accepted. For the computatio of the OC ad ASN of the SPRT, the largest latet root λ t, ρ) of the Pt) matrix has bee computed by fixig the values for t ad ρ. Figure 3 shows Table i j = = = =4 =47 l L= ij c ij =.62. Table 2: The value of c ij calculatig for SPRT. i j l L= ij c ij =.62. Table 3 i j = = = =4 =96 l L= ij c ij = the graphs of λ t, ρ) which have bee plotted agaist the values of t for differet values of ρ. AscabeseefromFigure 3, there exists oe ad oly oe real t =such that λ t )=. The derivative of λt, ρ) at t=has bee computed. The OC ad ASN fuctios are the evaluated usig the expressios betwee 3) ad7). The results of the OC ad ASN fuctios for testig H :ρ =.5 agaist H :ρ =.8 are give i Tables 4 ad 5 for α=.5, β =.5 ad ve α =.5, β =.,respectively. Figures 4 ad 5 give the OC curve ad the ASN curve for two differet error probabilities i the same figure. I this example, we wish to maitai ρ at the level.5 ad we wish to detect whether its value has icreased. The, the hypothesis test is H :ρ =.5agaist alterative H : ρ =.8. We accepted H hypothesis. There is o eed for ay chage i the system. So traffic itesity ρ) is the desired level. If ay chage i ρ, the queueig system would eed to be orgaized. Namely, if ρ has shifted to ρ from ρ ρ >ρ ), thecotrolactiocouldbetoicreasethemea or add a extra server. If ρ has shifted to ρ from ρ ρ <ρ ), the cotrol actiocouldbetodecreasethemeaor reduce the umber of servers. We ca uderstad from Table 4 that the umber of samples required for this hypothesis to be accepted is

8 8 Mathematical Problems i Egieerig λt, ρ) ρ =.5 ρ =.7 ρ =.6 ρ =.8 Figure 3: Graph of λt, ρ) agaist t for testig H :ρ =.5, H : ρ =.8. Lρ) α =.5, β =.5 α =.5, β =. Figure 4: The OC curve for testig H :ρ =.5 agaist H :ρ =.8. Table 4: The OC ad ASN values for α =.5 ad β =.5. ρ t ρ) λ, ρ) Lρ) E, ρ) ad the umber of samples required for it to be rejected is t ρ E, ρ) α =.5, β =.5 α =.5, β =. Figure 5: The ASN curve for testig H :ρ =.5 agaist H :ρ =.8. Table 5: The OC ad ASN values for α =.5 ad β =.. ρ tρ) λ, ρ) Lρ) E, ρ) ρ As ca be uderstood from the tables ad Figures 4 ad 5, the smaller the error probability eeded, the more the samplig umber ad power of a test are decreased. 6. Coclusios The objective of the cotrol techique is to detect chages i the traffic itesity ρ as quickly as possible, the take appropriate corrective actio, ad determie how much of a sample size is eeded i the applicatios. Thus, the sequetial probability ratio test provides a savig of up to fifty percet i the sample size accordig to traditioal methods. At the same time,theuseofsprtiseasyforobservigolytheumber of customers i the system at successive departure epochs Q, which are embedded Markov poits. I this paper, SPRT is theoretically ivestigated for two differet phase-type queueig systems which cosist of Hyperexpoetial ad Mixed Erlag. Also the ecessary coefficiets have bee obtaied for SPRT. I the applicatio part, the queue legths have bee geerated radomly for differet arrival rates λ), differet service rates ), ad selected system capacities N) by usig MATLAB 7.. R2a) programmig. Traffic itesities have also bee calculated. With the obtaied data ad predetermied α ad β, a simple hypothesis has bee established. Acceptig or rejectig the hypothesis has bee examied by

9 Mathematical Problems i Egieerig 9 SPRT. Afterward, the largest latet root λt, ρ) of the Pt) matrixhasbeecomputedbyfixigthevaluesfort ad ρ. Their graphs have bee draw. It has bee foud that there exists oe ad oly oe real t =such that λ t )=.The OC ad ASN have bee calculated with obtaied values, ad their graphs have bee draw by Microsoft Office Excel 27 programmig. As a result, the legth queue observed at the departure poits ca be maximum N,whereρ>,adthelegthqueue observed at the departure poits ca be miimum N,where ρ<. Accordig to the give α ad β, differeces i sample size have bee observed. It has bee observed that whe the error probability decreased the sample size grows ad the power of test β) icreases. Here, the value of eeded to calculate the traffic itesity was take from the mea 33). If you recall, the purpose of a cotrol techique is to detect chages i the traffic itesity ρ. Wheρ has shifted to ρ from the desig level ρ ρ < ρ ), a appropriate actioistaketobrigρ back to the desig level ρ.ithe same way, whe ρ has shifted to ρ from the desig level ρ ρ >ρ ), a appropriate actio is take to brig ρ back to the desig level ρ. Cosequetly, the cotrol actio could be to icrease mea or decrease mea for the phase-type queueig systems. Refereces [] R. M. Phatarfod, Sequetial aalysis of depedet observatios-i, Biometrika, vol. 52, pp , 965. [2] U. N. Bhat ad S. S. Rao, A statistical techique for the cotrol of traffic itesityi the queuig systemsm/g/ ad GI/M/, Operatios Research,vol.2,pp ,972. [3] U. N. Bhat, A statistical techique for the cotrol of traffic itesity i Markovia queue, Aals of Operatios Research, vol. 8, pp. 5 64, 987. [4] U. N. Bhat ad S. S. Rao, Statistical aalysis of queueig systems, Queueig Systems, vol., o. 3, pp , 987. [5] K. Harishchadra ad S. S. Rao, Statistical iferece about the traffic itesity parameter of M/E k / ad E k /M/ queues, Tech. Rep., Idia Istitute of Maagemet, Bagalore, Idia, 984. [6] S.S.Rao,U.N.Bhat,adK.Harishchadra, Cotroloftraffic itesity i a queue a method based o SPRT, Opsearch,vol. 2, o. 2, pp. 63 8, 984. [7] S. Jai ad J. G. C. Templeto, Problem of statistical iferece to cotrol the traffic itesity, Sequetial Aalysis, vol. 8, o. 2, pp , 989. [8] S. Jai, A autoregressive process ad its applicatio to queueig mode, Metro-Iteratioal Joural of Statistics,vol.58, o. -2, pp. 3 38, 2. [9] A. Wald, Sequetial Aalysis,Dover,NewYork,NY,USA,24. [] S. S. Rao, U. N. Bhat, ad K. Harishchara, OC ad ASN of the SPRT for a fiite markov chai ad a applicatio to the cotrol of queues, Tech. Rep. 3, Idia Istitute of Maagemet, Bagalore, Idia, 98. [] W. J. Stewart, Probability, Markov Chais, Queues, ad Simulatio, Priceto Uiversity Press, Priceto, NJ, USA, 29.

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Queuing Theory. Basic properties, Markovian models, Networks of queues, General service time distributions, Finite source models, Multiserver queues

Queuing Theory. Basic properties, Markovian models, Networks of queues, General service time distributions, Finite source models, Multiserver queues Queuig Theory Basic properties, Markovia models, Networks of queues, Geeral service time distributios, Fiite source models, Multiserver queues Chapter 8 Kedall s Notatio for Queuig Systems A/B/X/Y/Z: A