Analysis of DAR(1)/D/s Queue with Quasi-Negative Binomial-II as Marginal Distribution

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1 Applied Mathematic,,, doi:.436/am..96 Publihed Olie September ( Aalyi of DAR()/D/ Queue with Quai-Negative Biomial-II a Margial Ditributio Abtract Kaichukattu Korakutty Joe, Bidu Abraham Departmet of Statitic, St. Thoma College Palai Mahatma Gadhi Uiverity, Kottayam, Idia Departmet of Statitic, Baelio Pouloe II Catholicoe College, Mahatma Gadhi Uiverity, Kottayam, Idia kktc@gmail.com, babpc@rediffmail.com Received March 5, ; revied Jue 3, ; accepted Jue, I thi paper we coider the arrival proce of a multierver queue govered by a dicrete autoregreive proce of order [DAR()] with Quai-Negative Biomial Ditributio-II a the margial ditributio. Thi dicrete time multierver queueig ytem with autoregreive arrival i more uitable for modelig the Aychroou Trafer Mode(ATM) multiplexer queue with Variable Bit Rate (VBR) coded telecoferece traffic. DAR() i decribed by a few parameter ad it i eay to match the probability ditributio ad the decay rate of the autocorrelatio fuctio with thoe of meaured real traffic. For thi queueig ytem we obtaied the tatioary ditributio of the ytem ize ad the waitig time ditributio of a arbitrary packet with the help of matrix aalytic method ad the theory of Markov regeerative procee. Alo we coider egative biomial ditributio, geeralized Poio ditributio, Borel-Taer ditributio defied by Frak ad Melvi(96) ad zero trucated geeralized Poio ditributio a the pecial cae of Quai-Negative Biomial Ditributio-II. Fially, we developed computer programme for the imulatio ad empirical tudy of the effect of autocorrelatio fuctio of iput traffic o the tatioary ditributio of the ytem ize a well a waitig time of a arbitrary packet. The model i applied to a real data of umber of cutomer waitig for checkout i a airport ad it i etablihed that the model well uit thi data. Keyword: Dicrete Autoregreive Proce of Order [DAR()], Multierver ATM Multiplexer, Matrix Aalytic Method, Markov Reewal Proce, Markov Regeerative Theory, Telecoferece Traffic, Quai-Negative Biomial Ditributio-II, Geeralized Poio Ditributio, Borel-Taer Ditributio. Itroductio I B-ISDN/ATM etwork, IP packet or cell of voice, video, data are et over a commo tramiio chael o tatitical multiplexig bai. The performace aalyi of tatitical multiplexer whoe iput coit of a uperpoitio of everal packetized ource i ot a traightforward oe. The difficulty i modelig thi type of traffic i due to the correlated tructure of arrival. A commo approach i to approximate thi complex o reewal iput proce by aalytically tractable arrival proce, amely dicrete autoregreive proce (DAR). The impact of autocorrelatio i traffic procee o queueig performace meaure uch a mea queue legth, mea waitig time ad lo probabilitie i fiite buffer, ca be very dramatic. The DAR proce, cotructed ad aalyzed by Jacob ad Lewi [] ha developed ito oe of everal tadard tool for modellig iput traffic i telecommuicatio etwork. The dicrete autoregreive proce of order [DAR()] i kow to be a good model for VBR coded telecoferece traffic a i Elwalid et al. []. Kamou ad Ali [3] modeled a ATM multiplexer a a dicrete time multierver queueig ytem with o-off ource ad tudied the traiet ad tatioary ditributio of the umber of packet i the ytem. Hwag ad Sohraby [4] obtaied the cloed form expreio for the tatioary probability geeratig fuctio of the ytem ize of the dicrete time igle erver queue with DAR() iput. Hwag et al. [5] obtaied the waitig time ditributio of the dicrete time igle erver queue with DAR() iput. Choi ad Kim [6] aalyzed a

2 6 K. K. JOSE ET AL. multierver queue fed by DAR() iput. Kim et.al [7] derived mea queue ize i a queue with dicrete autoregreive arrival of order p. I thi paper we aalyzed a dicrete-time multi-erver queue with erver ( > ) havig determiitic ervice time (pecifically, ervice time i ) ad the followig arrival proce: Let A m be the umber of arrival at time m,,,. The A m = A m with probability β; otherwie, A m i ampled idepedetly from a quai-egative biomial ditributio-ii. The tatioary ditributio of the waitig time i that queue i calculated umerically with a matrix aalytic method. Specifically, the arrival proce i firt aalyzed at embedded time whe A m i ampled idepedetly of A m or whe A m i le tha the umber of erver. Thi aalyi reduce to a aalyi of a Markov chai of M/G/ type a preeted i Neut [8]. The the tatioary ditributio of A m at geeral m i derived, which i tur give the tatioary ditributio of the waitig time. The ret of the paper i arraged a follow. Quai- Negative Biomial Ditributio-II i decribed i Sectio. Queue with iput traffic a DAR() with margial Quai-Negative Biomial-II i explaied i Sectio 3. Aalyi of DAR() /D/ queue with margial Quai- Negative Biomial Ditributio-II i give i Sectio 4. The tatioary ditributio of the Markov reewal proce i give i Sectio 5. Derivig the tatioary ditributio of ytem ize ad waitig time of a arbitrary packet i explaied i Sectio 6 ad 7. The quatitative effect of the tatioary ditributio of ytem ize ad waitig time o the autocorrelatio fuctio a well a the parameter of the iput traffic i illutrated umerically i Sectio 8. The applicatio to real data et i give i Sectio 9.. Quai Negative Biomial Ditributio-II The quai-egative biomial ditributio (QNBD) obtaied by Jaarda [9], Se ad Jai [] ha the probability ma fuctio a P x;, p, p = x! ppxp!! x p xp x x for pxp ; >, p >, if p > ad p xp ; if p < ; x =,, where x be the umber of occurrece. Whe p = the QNBD reduce to egative biomial ditributio (NBD) ad whe =, QNBD reduce to quai geometric ditributio (QGSD) for =. QNBD ted to the Coul ad Jai [] geeralized Poio ditributio. But ufortuately the momet of thi ditributio appear i a ifiite erie which i ot uitable for ummatio. The () method of momet fail to provide quick etimate of it parameter. Hece Ahmad et al. [] itroduced a ew model of quai egative biomial ditributio-ii (QNBD- II). Thi ew model ha the probability ma fuctio x x p pxp p p p p xp px x x for < p < ad < p < ; x =,, Whe p thi ew model reduce to egative biomial ditributio. The probabilitie of QNBD-II decreae with the ucceive occurrece. Thi tedecy of probabilitie ugget it poible applicatio i reliability, biometry, ad urvival aalyi. The QNBD-II i ui-model ad oly it firt momet (mea) appear i compact form. The lower ad upper boud of Mode M i p p < M <, p <. p p p Mea =, p < p.. Remark ) Let X be a quai-egative biomial variate with parameter (, p, p) ad pmf give by (). If uch that p = ad p = the the radom variable X ted to geeralized Poio ditributio with parameter,. ) Let X be a quai-egative biomial variate with parameter (, p, p ) ad probability ma fuctio i (pmf) i give by (). If uch that = where = p the the radom variable X ted to Borel-Taer ditributio defied by Frak ad Melvi [3] 3) Let X be a quai-egative biomial variate with parameter (, p, p ) ad pmf give by (), the zerotrucated quai-egative-biomial ditributio-ii ted to zero-trucated geeralized Poio ditributio a. () 3. Queue with DAR() Arrival with Quai Negative Biomial Ditributio-II a Margial The iput ATM multiplexer with VBR coded telecoferece traffic i aumed to be DAR() with quaiegative biomial ditributio-ii a margial. Let Yt : t =,, be a equece of i.i.d radom variable. Y(t) aume poitive value oly ad bx = PY t = x, x =,,,. Whe the iput proce ha quai-egative biomial ditributio-ii a margial we have b x a the pmf of the form ().

3 K. K. JOSE ET AL. 6 Dicrete Autoregreive Proce of order (DAR() X t : t,,, i defied by the regreio equatio a X = Y X t = Z t X t Z t Y t, t =,, where Z t : t,,3, are i.i.d Beroulli radom variable with PZ= t = < ad PZ= t = ad Zt : t,,3, i aumed to be idepedet of Y t : t,,,. DAR() i determied by the parameter ad the ditributio b : x =,,, of Y(t), o that x X = Y X t with prob. X t = Y t with prob. The propertie of DAR() are a follow ) X t : t,, i tatioary ) The probability ditributio of X(t) i the ame a the ditributio of Y(t) PX t x bx x = =, =,, 3) The autocorrelatio fuctio for X(t) at lag t i obtaied a CovX ; X t) t t = =, t =,, Var X the parameter i the decay rate of the autocorrelatio fuctio. 4. Aalyi of DAR()/D/ Queue with Quai-Negative Biomial-II a Margial We aume that the iput proce i DAR() with quai-egative biomial ditributio-ii ditributio a the margial ditributio ad there are erver ( > ) whoe ervice occur at cotat rate. I thi iteger valued time queue, the time i divided ito lot of equal ize ad oe lot i eeded to erve a packet by a erver. We aume that packet arrival occur at the begiig of lot ad departure occur at the ed of the lot. Here X t : t,, repreet packet arrival o that X(t) i the umber of packet arrivig at the begiig of th the t lot. Let N(t) be the umber of packet i the ytem ay ytem ize, immediately before arrival at the begiig of the t th lot. The N t, X t : t,, i a two dimeioal Markov proce of M/G/ queue type. The tate pace i l, i E i =, =,,,,, The umber of phae i ifiity. So the computatio of tatioary ditributio of N t, X t : t,, i ot eay to work out. I practice by matrix aalytical method ad uig the theory Markov regeerative procee, we compute the tatioary ditributio of the ew proce at the embedded epoch t, =,, < t < t < t < t 3 a follow, we have t, = = if t > t - :Z t = or X t, =, Let J N X t = = N t, =,, J = Zt Zt if = =,,3 if = =,,3 The packet arrival at ad after t are idepedet of the iformatio prior to t give J. From thi, it i oberved that N, J : =,,, i the ew Markov reewal proce with tate pace,,, E The probability traitio matrix of the Markov reewal proce i computed a follow. ) For,, ad i,,, i, max i,, i with prob. max i,, with prob. ) For,, max i,, i with prob. bi i, i, with prob., bi ( ) i, mi,, with prob. bi g i= l, with prob. gll, l > where l i il if = = if g = b l i g = b, l =,, The traitio probability matrix P

4 6 B A A A B A A A B A Ac A A A A A A A A A3 A i obtaied a above. Where the elemetary matrice are i Ai = i bi bi i K. K. JOSE ET AL. Ai = i, g i i B = A,i i = We aume that the tability coditio = = EX t = mbm < i atified. m 5. The Statioary Ditributio of the Markov Reewal Proce Coider N J,, =,,, ad π i =limp N =, J = i,, i We apply matrix aalytic method a decribed below. The traitio probability matrix P ha ifiite order, o that it would have to be trucated before we implemet matrix aalytic method. We aume that there exit ome idex N uch that = for all > N. That i A N we aume that the Markov chai doe ot ump more tha N tep at a time o that the matrix i of fiite order, ee Latouche ad Ramawamy [4]. For a umerical illutratio, coider the cae whe = 5 ad N = 4. The the traitio probability matrix P ca be obtaied a B5 A6 A7 A8 A9 A A A A3 A4 A5 A6 A7 A8 A9 B4 A5 A6 A7 A8 A9 A A A A3 A4 A5 A6 A7 A 8 B3 A4 A5 A6 A7 A8 A9 A A A A3 A4 A5 A6 A7 B A3 A4 A5 A6 A7 A8 A9 A A A A3 A4 A5 A 6 B A A3 A4 A5 A6 A7 A8 A9 A A A A3 A4 A5 A A A A3 A4 A5 A6 A7 A8 A9 A A A A3 A 4 A A A A3 A4 A5 A6 A7 A8 A9 A A A A3 A A A A3 A4 A5 A6 A7 A8 A9 A A A A A A A3 A4 A5 A6 A7 A8 A9 A A A A A A3 A4 A5 A6 A7 A8 A9 A A A A A3 A4 A5 A6 A7 A8 A9 A A A A3 A4 A5 A6 A7 A 8 A A A A3 A4 A5 A6 A7 A A A A3 A4 A5 A 6 A A A A3 A4 A5 A A A A3 A4 A A A A3 A A A A A A

5 K. K. JOSE ET AL. 63 By arragig the traitio probability matrix ito (x) matrice we get Bˆ Bˆ Bˆ Aˆ ˆ ˆ A A P = Aˆ ˆ A Aˆ or equivaletly Bˆ Aˆ Aˆ 3 Aˆ ˆ ˆ A A P = Aˆ ˆ A Aˆ I geeral we ca ymbolize the traitio matrix P a Bˆ Bˆ Bˆ Bˆ Aˆ Aˆ Aˆ Aˆ P = Aˆ Aˆ Aˆ Aˆ * N = or equivaletly Bˆ Aˆ Aˆ 3 Aˆ Aˆ ˆ ˆ ˆ A A A P = Aˆ ˆ ˆ A A Aˆ The elemet of P ca be writte a B A A ˆ B A A B =, B A A A A A ˆ A A A A = A A A =,,, Bˆ = Aˆ, =,, A matrix P of the above tructure i aid to be of M/G/ type, which uderlie the imilarity to the embedded Markov chai of the M/G/ queue. With repect to the level, the Markov chai i called kip free to the left, ice i oe traitio the level ca be reduced oly by oe. By the matrix aalytic method we proceed a follow. Step : Fid the miimal oegative olutio G of the matrix equatio G = Aˆ G = G ca be give by the followig iteratio See Breuer [5] G = G = Aˆ k k k = = Gk k = ˆ G = A G, k =, 3, G G i a tochatic matrix,ad hece we ca top the iteratio procedure whe G. < reache where =.. From thi iteratio we obtaied the upper limit of k & let = k. From thi we come to kow the trucated idex N at which G become tochatic Step : Fid = ˆ H BG = ad a poitive row vector h atifyig hh = h Step 3: x = h = ˆ i ˆ i x x BiG xl A l i G i= l= i= ˆ i I Ai G, =, i= Step 4: Fially π, π π π =,,, Cx,, =,,,, where C = x = e ad e i the ( + ) dimeioal colum vector whoe compoet are all oe. 6. Statioary Ditributio of N t, X t, t =,, Oberve that N, J, t, =,, i a Markov

6 64 K. K. JOSE ET AL. reewal proce ad Ntt, X tt : t =,, give N, X, < t, N, J =, i tochatically equivalet to N t X t t give N, J =, i. Hece N t, X t : t =,, i, : =,, i a dicrete time Markov regeerative proce with the Markov reewal equece N, J, t : =,,.From the theorm See Kulkari [6] p =lim P N t, X t =,, =,, of t N t, X t : t =,, are determied by t πlie, =, N, =, N t X t J l i l= i= t= t p π liet t N, J = l, i l= i= We have t E N, = t= t N t, X t =, J l, i if i =,i ad = l b if i =, ad = l b if i =, = ad = l = l b if i =, >, l ad divide l otherwie. The umerator of Equatio (3) i π π b, πb, = i π i b,, i= We have Et t N, J = l, i if i, br br if i = r= r= The deomiator of Equatio (3) i πli πl br br l= i= l= r= r= where (3) (4) π = l l π l= l i the tatioary probability vector of the Markov procej : =,, whoe traitio probability matrix i P J = J = i i, (5) b b b br r= The ifiiteimal traitio matrix of (5) i Q = b b b r r = The balace equatio are Q = & e= By olvig the balace equatio we obtai the ta- tioary ditributio of the Markov proce J : =,, a bi, b r r= π li = (6) l =, i = b r r= By ubtitutig (6) ito (4) we obtai the deomiator of the right had ide of (3) a b r= Theorem 6. r The tatioary ditributio or the limitig probabilitie p = lim P N t, X t =,,, =,, are give by where p t π π b, πb, = i= br = r= i π b, i,

7 7. Statioary Ditributio of Waitig Time of a Arbitrary Packet K. K. JOSE ET AL. 65 Let W deote the waitig time of a arbitrary packet at teady tate. The for w =,,. P(W = w) = (Mea umber of arrival i a lot at teady tate whoe waitig time i w)/(mea umber of arrival i a lot) Suppoe that there are packet immediately before arrival at the begiig of the t th lot ad that the umber of packet arrival i at the begiig of the t th lot, o that N(t) = ad X(t) =. The the umber of packet whoe waitig time i w amog the oe who arrive at the begiig of the t th lot i mi w,, w< < w mi = w,, w = otherwie Therefore the mea umber of arrival i a lot at teady tate whoe waitig time w i w = = w w = w = p mi = w, p mi w, Sice the mea umber of arrival i a lot i, the followig theorem i obtaied from (7). Theorem 7. The ditributio of the waitig time W of a arbitrary packet i give by w PW w p w = = w mi =, w p mi w,, = w = 8. Empirical Study The complemetary ditributio fuctio of the tatioary ytem ize whe whe λ =.5 ad β =.3,.5,.7 &.9 ad the complemetary ditributio fuctio of the tatioary ytem ize whe β =.3 ad p =.9,.5,. &.4 (p =.64, 4,. &.4) repectively are derived. The parameter β give the iformatio o the tregth of correlatio of the iput proce. Statioary ytem ize i larger for the large β (ee Figure ). Alo tatioary ytem ize tochatically icreae whe the parameter p of the iput proce decreae (ee Figure ). The complemetary ditributio fuctio of the waitig time of a arbitrary packet,whe λ =.5 ad β =.3, Figure. Complemetary ditributio fuctio of the tatioary ytem ize, whe p =.45, p =.8, λ =.5. Figure. Complemetary ditributio fuctio of the tatioary ytem ize, whe λ =.5, β =.3..5,.7 &.9 ad the complemetary ditributio fuctio of the waitig time whe β =.3 ad p =.9,.5,. &.4 (p =.64, 4,. &.4) repectively are derived. Statioary waitig time of a arbitrary packet, i larger for large β (ee Figure 3). Alo tatioary waitig time of a arbitrary packet, tochatically icreae whe the iput parameter p decreae (ee Figure 4). We aume the umber of erver to be 3 Table -3 diplay the tatioary probabilitie of the ytem ize for differet value of p, p,&. Table 4 ad 5 diplay the tatioary probabilitie of waitig time of a arbitrary packet for differet value of p &.

8 66 K. K. JOSE ET AL. Figure 3. Complemetary ditributio fuctio of the waitig time of a arbitrary packet,whe p =.9, p =.64. Figure 4. Complemetary ditributio fuctio of the waitig time of a arbitrary packet, whe β =.3. Table. Showig the value of ditributio of tatioary ytem ize P(, ) for λ =.5, β =., p =.45, p =.8 ad = E E E E-4 5.E E-4 6.E E E-4 7.E E E E E-4 5.E E E E E-4 4.E E E - 4 Table. Showig the value of ditributio of tatioary ytem ize P(, ) for λ =.5, β =.3, p =.9, p =.94 ad = E - 4.E - 4.E - 4.E E - 5.E E E E E E - 5.E - 5.E E E E E - 5.E E - 5.E - 5..E - 5.E - 5.E E - 5.E E - 5.E E E E E E - 6.E E - 5.E - 5.E - 6.E E E E - 6.E E - 5.E - 5.E - 6.E E E E - 6.E E - 5.E E E - 6.E E E - 5.E - 6.E E - 6

9 K. K. JOSE ET AL. 67 Table 3. Showig the value of ditributio of tatioary ytem ize P(, ) for λ =.5, β =.3, p =.4, p =.4 ad = E E E E E E E E - 5..E E E E E E E E E - 5 Table 4. Showig the value of the ditributio of waitig time of a arbitrary packet P(W = ω) for differet value of β ad λ =.5, p =.9, p =.64 ad = 3. ω β Table 5. Showig the value of the ditributio of waitig time of a arbitrary packet P(W = ω) for differet value of p, β =.3, ad = 3. p ω p Aalyi ad Modelig of a Data Set I thi ectio we apply the model to a data o the umber of iitially waitig cutomer for checkig i a airport for a time period of 3 miute each t =,,,3. The data wa collected from morig 8. A.M to.3 P.M for oe week. Thi iclude all the buy period a well a idle period. The data i take from the file cutomer checkout.xlx available i [7]. Table 6 give the frequecy ditributio of the corre- podig data, where x i the umber of cutomer iitially waitig for the ervice. I the preet paper we aumed the umber of arrival a DAR() with margial Quai Negative Biomial II ditributio. Thu the data et ca be fitted to the the Quai Negative Biomial II ditributio a follow. To tet whether there i a igificat differece betwee a oberved ditributio ad the Quai Negative Biomial II ditributio, we ue Kolmogorov-Smirov [K.S.] tet for H : Quai Negative Biomial II ditri-

10 68 K. K. JOSE ET AL. Table 6. Table howig the frequecy ditributio of the umber of cutomer waitig for checkout. x frequecy x frequecy Total 336 butio with parameter p =. ad p =.53 i a good fit for the give data. Here the calculated value of the K.S. tet tatitic i.7857 ad the critical value correpodig to the igificace level. i.8894, howig that the aumptio for umber of arrival follow Quai Negative Biomial II ditributio i valid (ee Figure 5). By applyig matrix aalytic method we obtai the tatioary ditributio of ytem ize ad waitig time of a arbitrary cutomer for the Quai Negative Biomial II/D/ queue. Here the mea = λ =4.35. To atify the tability coditio we aume the umber of erver a =5. Alo we aume the value of autocorrelatio fuctio =., p =. ad p =.53. Table 7 ad 8 diplay the tatioary ditributio of waitig time of a arbitrary cutomer ad ytem ize. Figure 5. The Probability hitogram of real data ad the Quai Negative Biomial II ditributio with p =. ad p =.53. Table 7. Table howig the tatioary ditributio of waitig time of a arbitrary cutomer P(W =ω)whe β =., λ = 4.35, = 5. w p(w) Table 8. Table howig the tatioary ditributio of ytem ize P(, ) whe β =., λ = 4.35, =

11 K. K. JOSE ET AL. 69. Cocluio I thi paper we aalyze DAR()/D/ queue with Quai- Negative Biomial Ditributio-II a the margial ditributio. Baed o the matrix aalytic method ad by uig the theory of Markov regeerative procee, we obtaied the tatioary ditributio of the ytem ize ad the waitig time of a arbitrary packet. From the defiitio of autocorrelatio fuctio we ca ay that the larger the parameter β, the lower the decay of the autocorrelatio of the iput proce. So it i expected that tatioary ytem ize ad waitig time for the cae of large β are tochatically larger tha thoe for the cae of mall β. Alo the tatioary ytem ize ad waitig time icreae whe the iput parameter decreae.. Referece p [] P. A. Jacob ad P. A.W. Lewi, Dicrete Time Serie geerated by Mixture III: Autoregreive Procee (DAR(p)), Naval Potgraduate School, Moterey, 978. [] A. Elwalid, D. Heyma, T. V. Lakma, D. Mitra ad A. Wei, Fudametal Boud ad Approximatio for ATM Multiplexer with Applicatio to Video Telecoferecig, IEEE Joural of Selected Area i Commuicatio, Vol. 3, No. 6, 995, pp doi:.9/ [3] F. Kamou ad M. M. Ali, A New Theortical Approach for the Traiet ad Steady State Aalyi of Multierver ATM Multiplexer with Correlated Arrival, 995 IEEE Iteratioal Coferece o Commuicatio, Vol., 995, pp doi:.9/icc [4] G. U. Hwag ad K. Sohraby, O the Exact Aalyi of a Dicrete Time Queueig Sytem with Autoregreive Iput, Queueig Sytem, Vol. 43, No. -, 3, pp doi:.3/a: [5] G. U. Hwag, B. D. Choi ad J. K. Kim, The Waitig Time Aalyi of a Dicrete Time Queue with Arrival a a Autoregreive Proce of Order, Joural of Applied Probability, Vol. 39, No. 3, 3, pp [6] B. D. Choi, B. Kim, G. U. Hwag ad J. K. Kim, The Aalyi of a Multierver Queue Fed by a Dicrete Autoregreiive Proce of Order, Operatio Reearch Letter, Vol. 3, No., 4, pp doi:.6/s (3)68-3 [7] J. Kim, B. Kim ad K. Sohraby, Mea Queue Size i a Queue with Dicrete Autoregreive Arrival of Order p, Aal of Operatio Reearch, Vol. 6, No., 8, pp doi:.7/ [8] M. F. Neut, Structured Stochatic Matrice of the M/G/ Type ad Their Applicatio, Dekker, New York [9] K. G. Jaardha, Markov-Polya ur Model with Pre- Determied Strategie, Guarat Statitical Review, Vol., No., 975, pp [] K. Se ad R. Jai Geeralized Markov-Polya ur Model Withpre-Determied Strategie, Joural of Statitical Plaig ad Iferece, Vol. 54, 996, pp doi:.6/ (95)6- [] G. C. Jai ad P. C. Coul, A Geeralized Negative Biomial Ditributio, Society for Idutrial Mathematic, Vol., No. 4, 97, pp doi:.37/56 [] S. B. Ahmad, A. Haa ad M. J. Iqbal, O a Quai Negative Biomial Ditributio II ad It Applicatio, Preprit,. [3] A. H. Frak ad A. B. Melvi, The Borel-Taerditributio, Biometrica, Vol. 47, No. -, 96, pp [4] G. Latouche ad V. Ramawamy, Itroductio to Matrix Aalytical Method i Stochatic Modellig, Society for Idutrial Mathematic, Peylvaia, 99. [5] L. Breuer ad D. Baum A Itroductio to Queueig Theory ad Matrix Aalytic Method, Spriger, Berli, 4. [6] V. G. Kulkari Modellig ad Aalyi of Stochatic Sytem, Chapma & Hall, Lodo, 995. [7] Cutomer Check Out. xlx