Performance evaluation of two Markovian retrial queueing model with balking and feedback

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1 Acta Unv. Sapentae, Mathematca, 5, 2 (2013) Performance evaluaton of two Markovan retral queueng model wth balkng and feedback A. A. Bouchentouf Djllal Labes Unversty of Sd Bel Abbes Department of Mathematcs B. P. 89, Sd Bel Abbes emal: bouchentouf amna@yahoo.fr. Belarb Djllal Labes Unversty of Sd Bel Abbes Department of Mathematcs B. P. 89, Sd Bel Abbes emal: faza belarb@yahoo.fr Abstract. In ths paper, we consder the performance evaluaton of two retral queueng system. Customers arrve to the system, f upon arrval, the queue s full, the new arrvng customers ether move nto one of the orbts, from whch they make a new attempts to reach the prmary queue, untl they fnd the server dle or balk and leave the system, these later, and after gettng a servce may comeback to the system requrng another servce. So, we derve for ths system, the jont dstrbuton of the server state and retral queue lengths. Then, we gve some numercal results that clarfy the relatonshp between the retrals, arrvals, balkng rates, and the retral queue length. 1 Introducton In the parlance of queueng theory, such a mechansm n whch ejected (or rejected) customers return at random ntervals untl they receve servce s called a retral queue. Retral queues have an mportant applcaton n a wde varety of felds, they are lkewse prevalent n the evaluaton and desgn of 2010 Mathematcs Subject Classfcaton: 60K25, 68M20, 90B22 Key words and phrases: queueng models, retral queues, balkng, jont dstrbuton functon, confluent hypergeometrc functons 132

2 Performance evaluaton of two Markovan retral queueng model 133 computer networks as they are n telecommuncatons, computer networks, and partcularly wreless networks. A retral queue s smlar to any ordnary queueng system n that there s an arrval process and one or more servers. The fundamental dfferences are frstly, the enttes who enter durng a down or busy perod of the server or servers may reattempt servce at some random tme n the future, and secondly a watng room, whch s known as a prmary queue n the context of retral queues, s not mandatory. In place of the ordnary watng room s a buffer called an orbt to whch enttes proceed after an unsuccessful attempt at servce, and from whch they retry servce accordng to a gven probablstc or determnstc polcy. Owng to the utlty and nterestng mathematcal propertes of retral queueng models, a vast lterature on the subject has emerged over the past several decades. or a general survey of retral queues and a summary of many results, the reader s drected to the works of [6, 8, 7, 5, 12, 15] and references theren. In [4] Cho and Km consdered the M/M/c retral queues wth geometrc loss and feedback when c = 1, 2, they found the jont generatng functon of the number of busy servers and the queue length by solvng Kummer dfferental equaton for c = 1, and by the method of seres soluton for c = 1, 2. Retral queueng model MMAP/M 2 /1 wth two orbts was studed by Avrachenkov, Dudn and Klmenok [3], n ther paper, authors consdered a retral sngleserver queueng model wth two types of customers. In case of the server occupancy at the arrval epoch, the customer moves to the orbt dependng on the type of the customer. One orbt s nfnte whle the second one s a fnte. Jont dstrbuton of the number of customers n the orbts and some performance measures are computed. An M/M/1 queue wth customers balkng was proposed by Haght [9], Sumeet Kumar Sharma [10] studed the M/M/1/N queung system wth retenton of reneged customers, Kumar and Sharma [11] studed a sngle server queueng system wth retenton of reneged customers and balkng. Kumar and Sharma [14] consder a sngle server, fnte capacty Markovan feedback queue wth balkng, balkng and retenton of reneged customers n whch the nter-arrval and servce tmes follow exponental dstrbuton. In our paper, we consder a retral queueng model wth two orbts O 1 and O 2, balkng and feedback. In case of the server occupancy at the arrval epoch, the arrvng customers have to choose between the two orbts dependng on ther thresholds f they decde to stay for an attempt to get served or leave the system (balk), and after gettng a servce, customers may comeback to the system requrng another servce. The man result n ths work conssts n dervng the approxmate analyss of the system.

3 134 A. A. Bouchentouf,. Belarb 2 Mathematcal model gure 1: Retral queueng model wth balkng and feedback We consder a retral queueng model wth two orbts O 1 and O 2, new customers arrve from outsde to the servce node accordng to a posson process wth rate λ. If the queue s not full upon prmary call arrvals, then the customers wat n the queue, thus wll be served accordng to the IO order, where servce tmes B(t) are assumed to be ndependent and exponentally dstrbuted wth mean 1/µ. However, f upon arrval, the customers fnd the queue full, then they decde to stay for an attempt to get served wth probablty β = 1 β or leave the system wth probablty β, 0 β 1. The arrvng customers who decde to stay for an attempt, they have to choose one of the orbts O 1, O 2 ; dependng on ther thresholds; f the number of customers n orbt O 1 s qute larger than that of orbt O 2, the customer wll move nto the orbt O 2 wth probablty ββ 2 ; 0 β 2 1, otherwse he/she removes nto orbt O 1 wth probablty ββ 1 ; 0 β 1 1. Notce that f the threshold of customers n orbt O 1 s qute larger than that of orbt O 2, the customers n orbt O 1 wll make the attempts frstly and vce versa. Afterward, customers go n the retral queues and make attempts

4 Performance evaluaton of two Markovan retral queueng model 135 to reach the prmary queue, where the attempt tmes are assumed also to be ndependent and exponentally dstrbuted wth mean 1/α, = 1, 2. nally, after the customer s served completely, he/she may decde ether to jon the retral groups O 1 or O 2 agan for another servce wth probablty ξδ 1 ; (δ 1 s the probablty that the customer chooses orbt O 1 ), wth 0 δ 1 1, or ξδ 2 ; (δ 2 s the probablty that the customer chooses orbt O 2 ), wth 0 δ 2 1, or leaves the system forever wth probablty ξ, 0 ξ 1. Ths sort of system abstracts and characterzes dfferent practcal stuatons n the telecommuncaton networks. or example, the mechansm based automatc repeat request protocol n data transmsson systems may be modeled by a retral queue system wth feedback, snce lost packets are negatvely acknowledged by the recevers, then the senders send them once agan. In ths paper we provde approxmate expressons for our queueng performance measures; we nvestgate the jont dstrbuton of the server state and queue length under the system steady state assumpton. The condton of system stablty s assumed to be hold, urther analyss around the stablty of retral queues can be found n [2], where E. Altman and A. A. Borovkov provded the general condtons under whch ρ (system s load) < 1 s a suffcent condton for the stablty of retral queung systems. 3 Man result Theorem 1 or our retral queueng model wth two orbts, balkng and feedback n the steady state: 1. The average of the queue length along the dle perod of the server s expressed by ( ( β ββ (λ + α ) + E(N, S = 0) = m ξδ ) µ ββ β(λ + α ) + µ(1 δ ξ) α β + µ(1 δ ξ) λ ββ { β β (λ + 2α ) + ξδ µ, β(2α + λ) + µ(1 δ ξ) ( α β 3 ) + ββ α βα α β λ β 2 β 2 ( ββ (λ + α ) + ξδ )( µ ββ (λ + 2α ) + ξδ ) µ β(λ + α ) + µ(1 δ ξ) β(λ + 2α ) + µ(1 δ ξ) { β β(λ + α ) + ββ ξδ µ, ββ (α + λ) + µδ ξ ). ββ α ββ α α β.

5 136 A. A. Bouchentouf,. Belarb 2. The average of the queue length along the busy perod of the server s expressed by ( ( β ββ (λ + α ) + E(N, S = 1) = m ξδ ) µ ββ β(λ + α ) + µ(1 δ ξ) { β β (λ + 2α ) + ξδ µ, β(2α + λ) + µ(1 δ ξ) ). ββ α βα α β 3. The average of the queue length s gven by 2 ( ( β ββ (λ + α ) + E(N, S = 0) + E(N, S = 1) = m ξδ ) µ ββ =1 β(λ + α ) + µ(1 δ ξ) α β + µ(1 δ ξ) + 1 λ ββ { β β (λ + 2α ) + ξδ µ, β(2α + λ) + µ(1 δ ξ) ββ α βα α β ( α β 3 )( ββ (λ + α ) + + ξδ ) µ λ β 2 β 2 β(λ + α ) + µ(1 δ ξ) ( ββ (λ + 2α ) + ξδ ) µ β(λ + 2α ) + µ(1 δ ξ) { β β (λ + 3α ) + ξδ µ, β(3α + λ) + µ(1 δ ξ) ββ α βα α β { β β(λ + α ) + ββ ξδ µ, ββ (α + λ) + µδ ξ ). ββ α ββ α α β Proof. To prove the theorem, we should frstly ntroduce the system statstcal equlbrum equatons for the system, so let us denote N 1 (t), N 2 (t) the number of repeated calls n the the retral queue O 1 respectvely O 2 at tme t, and S(t) represents the server state, where t takes two values 1 or 0 at tme t when the server s busy or dle respectvely. Thus, a process {S(t), N 1 (t), N 2 (t)} whch descrbes the number of customers n the system s the smplest and smultaneously the most mportant process assocated wth the retral queueng system descrbed n g.1. To smplfy our analyss, we suppose that the servce tme functon B(t) s exponentally dstrbuted. Thus, {S(t), N 1 (t), N 2 (t)} forms a markov process, where we can consder the markov chan of ths process representng ths system s embedded at jump customers arrval tmes rather than a chan embedded at servce completon epochs. Hence, the process {(S(t), N 1 (t), N 2 (t)) : t

6 Performance evaluaton of two Markovan retral queueng model 137 0} forms a Markov chan wth a state space {0, 1} {0, 1,..., N 1 } {0, 1,..., N 2 }, where {S, N 1, N 2 } lm t {S(t), N 1 (t), N 2 (t)} n the steady state. As a result, n the steady state the jont probabltes of server state S and the retral queue lengths N 1, N 2, P 0n1 n 2 = P{S = 0, N 1 = n 1, N 2 = n 2 }, and P 1n1 n 2 = P{S = 1, N 1 = n 1, N 2 = n 2 }, can be characterzed through the correspondng partal generatng functons for z 1 1, z 2 1 by P 0 (z 1 ) = n 1 =0 P 0n 1 n 2 z n 1 1, P 0(z 2 ) = n 2 =0 P 0n 1 n 2 z n 2 2 and P 1(z 1 ) = n 1 =0 P 1n 1 n 2 z n 1 1, P 1 (z 2 ) = n 2 =0 P 1n 1 n 2 z n 2 2. Consequently, we can descrbe the set of statstcal equlbrum equatons for these probabltes (P 0n1 n 2, P 1n1 n 2 ) as follows: (λ + n 1 α 1 )P 0n1 n 2 = ξµp 1n1 n 2 + ξδ 1 µp 1n1 1n 2 (1) (λββ 1 + µ + n 1 βα 1 )P 1n1 n 2 = ββ 1 λp 1n1 1n 2 + (n 1 + 1)βα 1 P 1n1 +1n 2 + (n 1 + 1)α 1 P 0n1 +1n 2 + λp 0n1 n 2 (2) (λ + n 2 α 2 )P 0n1 n 2 = ξµp 1n1 n 2 + ξδ 2 µp 1n1 n 2 1 (3) (λ ββ 2 + µ + n 2 βα 2 )P 1n1 n 2 = ββ 2 λp 1n1 n (n 2 + 1)βα 2 P 1n1 n (n 2 + 1)α 2 P 0n1 n λp 0n1 n 2. (4) Now to contnue n dervng the jont dstrbuton, we multply the equatons (1), (2), (3) and (4) by =n z n, = 1, 2 whch yelds to the followng equatons : λp 0 (z ) + α z P 0 (z ) = ξµp 1 (z ) + ξδ µz P 1 (z ) (5) [ λ ββ (1 z ) + µ) ] P 1 (z ) + α β(z 1)P 1 (z ) = α P 0 (z ) + λp 0 (z ). (6) By takng the sum of equaton (5) and (6), then dvde the sum by (z 1) we obtan α P 0 (z ) + α βp 1 (z ) = ( ββ λ + ξδ µ)p 1 (z ). (7) By substtutng equaton (7) nto (6), we can express P 0 (z ) n terms of P 1 (z ), P 1 (z ) as follows: P 0 (z ) = ( α β λ )z P 1 (z ) + ( µ λ (1 δ ξ) ββ z ) P1 (z ). (8) By dfferentatng equaton (8), we get P 0 (z ) = α β λ z P 1 (z ) + ( µ(1 δ ξ)+α β λ ββ z ) P 1 (z ) ββ P 1 (z ). (9) By substtutng equatons (8) and (9) nto (5), we obtan a dfferental equaton of P 1 (z )

7 138 A. A. Bouchentouf,. Belarb ( µ(1 δ ξ) + (λ + z P 1 (z α )β ) + λ ββ ) α β α β z P 1 (z ) λ ( ββ (λ + α ) + (1 ξδ )µ ) α 2 β P 1 (z ) = 0. (10) Consequently, we transform the equaton (10) nto Kummer s dfferental equaton, snce t has already a soluton. Let Y(x ) = P 1 (z (x )) and z = βα ββ λ x, = 1, 2 whch transforms (10) nto x Y (x ) + (λ+α )β+µ(1 δ ξ) βα x Y (x ) The equaton (11) can be rewrtten as follows such that a = β β(α +λ)+µδ ξ α β β β β(α +λ)+µδ ξ Y α β β (x ) = 0. (11) x Y (x ) + (d x )Y (x) a Y(x ) = 0 (12) and d = (λ+α )β+µ(1 δ ξ) βα. Referrng to [1], [13], the equaton (12) has a regular sngular pont at x = 0, and an rregular sngularty at x =. urthermore, the soluton of equaton (12) s found analytcally n a unte crcle, U = {x : x 1} whch represents n turn the soluton of kummer s functon Y(x ) and expressed by Y(x ) = m (a ; d ; x ), m 0 so, equaton (10) s solved for P 1 (z ) as follows P 1 (z ) = m { ββ (λ + α ) + ξδ µ ββ α ; (λ + α )β + µ(1 δ ξ) ; ββ λ α β α β z }, z 1. (13) Referrng to [13], the frst dervatve of Kummer s functon (a ; d ; x ) s defned as follows: d dx = a d (a + 1; d + 1; x ), hence P 1 (z ) s expressed as follows: { ( β ββ P 1 (z (λ + α ) + ) = m ξδ ) { µ ββ (λ + 2α ) + ξδ µ ; ββ β(λ + α ) + µ(1 δ ξ) ββ α (λ + 2α )β + µ(1 δ ξ) ; ββ }} (14) λ α β α β z, z 1.

8 Performance evaluaton of two Markovan retral queueng model 139 Then we replace nto equaton (8) for P 0 (z ), P 1 (z ) and P 1 (z ) by ther equvalence n equatons (13) and (14), and hence P 0 (z ) s expressed as follows: [ α β 2 ( ββ (λ + α ) + P 0 (z ) = m ξδ ) µ λ ββ β(λ + α ) + µ(1 δ ξ) { β β (λ + 2α ) + ξδ µ, β(2α + λ) + µ(1 δ ξ) ββ α βα α β (15) µ(1 δ1 ξ) + λ ββ { β β(λ + α ) + ξδ µ, ββ (α + λ) + µδ ξ ] ββ α βα α β Then at the boundary condton, where z = 1, = 1, 2 we can ge the value of m through P 0 (1) + P 1 (1) = 1 [ α β 2 ( ββ (λ + α ) + m = ξδ ) µ λ ββ β(λ + α ) + µ(1 δ ξ) { β β (λ + 2α ) + ξδ µ, β(2α + λ) + µ(1 δ ξ) ββ α βα α β (16) µ(1 δ1 ξ) + λ ββ + 1 { β β(λ + α ) + ξδ µ, ββ (α + λ) + µδ ξ ] 1 ββ α βα α β So, the generatng functons of the jont dstrbuton of server state S and queue length N are gven by ( ββ (λ + α ) + ξδ µ ) β(λ + α ) + µ(1 δ ξ) [ P 0 (z ) = E(z N α β 2, S = 0) = m λ ββ { β β (λ + 2α ) + ξδ µ P 1 (z ) = E(z N, β(2α + λ) + µ(1 δ ξ) ββ α βα α β µ(1 δ1 ξ) + λ ββ { β β(λ + α ) + ξδ µ, ββ (α + λ) + µδ ξ ] ββ α βα α β z { ββ (λ + α ) + : S = 1) = m ξδ µ ; ββ α β(λ + α ) + (1 δ ξ)µ ; ββ } λ βα α β z, z 1.

9 140 A. A. Bouchentouf,. Belarb Consequently, the average of the queue length along the dle perod of the server s equvalent to P 0 (1), whch s expressed by ( ( β ββ (λ + α ) + E(N, S = 0) = m ξδ ) µ α β + µ(1 δ ξ) ββ β(λ + α ) + µ(1 δ ξ) λ ββ { β β (λ + 2α ) + ξδ µ, β(2α + λ) + µ(1 δ ξ) ( α β 3 ) + ββ α βα α β λ β 2 β 2 ( ββ (λ + α ) + ξδ ) ( µ ββ (λ + 2α ) + ξδ ) µ β(λ + α ) + µ(1 δ ξ) β(λ + 2α ) + µ(1 δ ξ) { β β (λ + 3α ) + ξδ µ, β(3α + λ) + µ(1 δ ξ) ββ α βα α β { β β(λ + α ) + ββ ξδ µ, ββ (α + λ) + µδ ξ ) ββ α ββ α α β And the average of the queue length along the busy perod of the server s equvalent to P 1 (1), whch s expressed by { ( E(z N β ββ (λ + α ) + : S = 1) = m ξδ ) µ ββ β(λ + α ) + µ(1 δ ξ) { ββ (λ + 2α ) + ξδ µ ; (λ + 2α )β + µ(1 δ ξ) ; ββ }} (17) λ ββ α α β α β Thus the average of the queue length n the retral queung system s the sum of P 0 (1) and P 1 (1), whch s gven by 2 ( ( β ββ (λ + α ) + E(N, S = 0) + E(N, S = 1) = m ξδ ) µ ββ =1 β(λ + α ) + µ(1 δ ξ) α β + µ(1 δ ξ) + 1 λ ββ { β β (λ + 2α ) + ξδ µ, β(2α + λ) + µ(1 δ ξ) ββ α βα α β ( α β 3 )( ββ (λ + α ) + + ξδ ) ( µ ββ (λ + 2α ) + ξδ ) µ λ β 2 β 2 β(λ + α ) + µ(1 δ ξ) β(λ + 2α ) + µ(1 δ ξ) { β β (λ + 3α ) + ξδ µ, β(3α + λ) + µ(1 δ ξ) ββ α βα α β { ββ β β(λ + α ) + ξδ µ, ββ (α + λ) + µδ ξ ) ββ α ββ α α β

10 Performance evaluaton of two Markovan retral queueng model Numercal results The average watng tme W n the steady state s often consdered to be the most mportant of performance measures n retral queung systems. However, W s an average over all prmary calls, ncludng those calls whch receve mmedate servce and really do not wat at all. A better grasp of understandng the watng tme process can be obtaned by studyng frst the relatonshp between the retral queue length E(N) = E(N 1 ) + E(N 2 ) and other nputs,outputs and feedback parameters. We have conducted some prelmnary analyss through some smulatons done on the queue lengths, n order to show the mpact of the dfferent parameters and ts relatonshp wth the retral queue length E(N). The prmary objectve behnd ths was to understand what does happen at some telecommuncaton systems where redals or connecton retrals arse naturally. These analyss nvolved three scenaros fgure 2-fgure 4 n order to clarfy the relatons n dfferent stuatons among the nput, output, balk and feedback parameters. These scenaros are realzed through smulatons va Matlab program. To begn, we chose a sgnfcant values for the parameters so as to meet the requrements of the phase-mergng algorthm. or the frst fgure, for each value of ξ ( ξ = 0; 0.2; 0.4; 0.6; 0.8; 1) selected, we vary µ from 0 to 1 n ncrements of 0.1, where we evaluate E(N) at dfferent values of servce completon probablty whle β 1 = β 2 = α 1 = α 2 = δ 1 = δ 2 = 0.5, β = 0.7, λ = 0.7. The numercal results are summarzed n the followng table: µ Average Retral Queue Length ξ = 1 ξ = 0.8 ξ = 0.6 ξ = 0.4 ξ = 0.2 ξ = 0 0 E(N,C = 0) + E(N,C = 1) E(N,C = 0) + E(N,C = 1) ,5577 2,4302 2,3026 2,1749 2, E(N,C = 0) + E(N,C = 1) ,7798 2,5434 2,3077 2,0726 1, E(N,C = 0) + E(N,C = 1) ,9723 2,6409 2,3119 1,9857 1, E(N,C = 0) + E(N,C = 1) ,1417 2,7258 2,3154 1,9111 1, E(N,C = 0) + E(N,C = 1) ,2926 2,8005 2,3183 1,8467 1, E(N,C = 0) + E(N,C = 1) ,4280 2,8670 2,3208 1,7905 1, E(N,C = 0) + E(N,C = 1) ,5506 2,9264 2,3229 1,7413 1, E(N,C = 0) + E(N,C = 1) ,6623 2,9800 2,3247 1,6977 1, E(N,C = 0) + E(N,C = 1) ,7645 3,0284 2,3262 1,6591 1, E(N,C = 0) + E(N,C = 1) ,8586 3,0726 2,3276 1,6245 0,9639 The frst fgure shows that along the ncrease of µ the retral queue lengths ncrease when the values of ξ become larger; for nstance when ξ = 1; 0.8; 0.6 and decrease when ξ become smaller; for nstance when ξ = 0; 0.2; 0.4. Obvously, ths refers to the possblty of acceptng repeated and prmary calls

11 142 A. A. Bouchentouf,. Belarb becomes large. Ths fgure shows us also that when ξ becomes greater than 0.6 or the feedback probablty becomes less than 0.6, then E(N) s not affected remarkably or t decreases very slowly. gure 2: Average retral queue length E(N) & servce server rate µ or the second fgure, for each value of β such that or β = 0, 5 we choose a sgnfcant parameters α 1 = α 2 = 0.7, δ 1 = 0.1, δ 2 = 0.9 and β 1 = β 2 = 0.5. or β = 0.7, we choose a sgnfcant parameters α 1 = α 2 = 0.7, δ 1 = 0.1, δ 2 = 0.9, and β 1 = 0.6, β 2 = 0.4, we vary ξ from 0 to 1 n ncrements of 0.1, where we evaluate E(N) at dfferent values of balkng probablty β, whle µ = 0.8 and λ = 0.7. The numercal results are summarzed n the followng table: ξ Average Retral Queue Length β = 0.3 β = E(N, C = 0) + E(N, C = 1) 4,8845 2, E(N, C = 0) + E(N, C = 1) 4,4893 2, E(N, C = 0) + E(N, C = 1) 4,1030 2, E(N, C = 0) + E(N, C = 1) 3,7245 2, E(N, C = 0) + E(N, C = 1) 3,3533 2, E(N, C = 0) + E(N, C = 1) 2,9887 1, E(N, C = 0) + E(N, C = 1) 2,6304 1, E(N, C = 0) + E(N, C = 1) 2,2782 1, E(N, C = 0) + E(N, C = 1) 1,9322 1, E(N, C = 0) + E(N, C = 1) 1,5926 1, E(N, C = 0) + E(N, C = 1) 1,2598 0,9294

12 Performance evaluaton of two Markovan retral queueng model 143 gure 3: Average retral queue length E(N) & probablty of servce completon ξ The second fgure shows that E(N) for our model wth balkng and feedback s not affected by feedback probablty ξ when the probablty β of non-balkng or returnng to retral group after customer attempt s falure becomes less than 0.5. However, E(N) ncreases rapdly as ξ and β become hgh. or the thrd fgure, or each value of α (α 1 = α 2 = 0.1 and α 1 = α 2 = 0.8) selected, we vary β from 0.1 to 0.9 n ncrements of 0.1, such that for a good requrement we choose for β = 0.1 β 1 = 0.7 β 2 = 0.3 for β = 0.2 β 1 = 0.9 β 2 = 0.1 for β = 0.3 β 1 = 0.95 β 2 = 0.05 for β = 0.4 β 1 = 0.97 β 2 = 0.03 for β = 0.5 β 1 = 0.98 β 2 = 0.02 for β = 0.6 β 1 = 0.99 β 2 = 0.01 for β = 0.7 β 1 = β 2 = for β = 0.8 β 1 = β 2 = for β = 0.9 β 1 = β 2 = Then, we evaluate E(N) at dfferent values of retral probablty α, whle δ 1 = δ 2 = 0.5, ξ = 0.5, µ = 0.8 and λ = 0.7. The numercal results are summarzed n the followng table:

13 144 A. A. Bouchentouf,. Belarb β Average Retral Queue Length α 1 = α 2 = 0.1 α 1 = α 2 = E(N, C = 0) + E(N, C = 1) , E(N, C = 0) + E(N, C = 1) 7,8458 6, E(N, C = 0) + E(N, C = 1) 8,9262 6, E(N, C = 0) + E(N, C = 1) 9,7914 7, E(N, C = 0) + E(N, C = 1) 10,2588 7, E(N, C = 0) + E(N, C = 1) 14, , E(N, C = 0) + E(N, C = 1) 14, , E(N, C = 0) + E(N, C = 1) 15, , E(N, C = 0) + E(N, C = 1) 16, ,9861 gure 4: Average retral queue length E(N) & non-balkng rate β gure 4 shows that along the desgn of retral queung system, we have to assgn equvalent values for the non-balkng probablty β and the retral probablty α n order to keep the retral queue length as short as possble. Ths can be concluded from the fgure snce when α takes values greater or equal to 0.5, and β gets values less than 0.5 E(N) becomes small. As a concluson, we conclude that gures 2 through 4 ndcate that the phase-mergng algorthm s reasonably effectve n approxmatng E(N), for all values of µ, ξ, β, and α.

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