The Performance Analysis Of A M/M/2/2+1 Retrial Queue With Unreliable Server

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1 Journal of Statstcal Scence and Applcaton, October 5, Vol. 3, No. 9-, do:.765/38-4/5.9.3 D DAV I D PUBLISHING The Performance Analyss Of A M/M//+ Retral Queue Wth Unrelable Server R. Kalyanaraman and M. Seenvasan Department of Mathematcs, Annamala Unversty, Annamalanagar-68, Inda The M / M / r/ r+ d retral queung system wth unrelable server s consdered. The customers arrve accordng to a Posson process and the servce tme dstrbuton s negatve exponental. The lfe tme of the server and repar tmes are also negatve exponental. If the system s full at the tme of arrval of a customer, the customer enters nto an orbt. From the orbt the customer tres hs luck. The tme between two successve retral follows negatve exponental dstrbuton. The model s analyzed usng Matrx Geometrc Method. The ont dstrbuton of system sze and orbt sze n steady state s studed. Some system performance measures are obtaned. We also provde numercal examples by takng partcular values to the parameters. Keywords: Retral queue, Matrx-Geometrc Method, Retral rate, Statonary dstrbuton, Performance measure. Introducton In a queung system, f customers arrve for servce and fnd all servers are busy, leave the servce area temporarly, stay n orbt and repeat ther demand after a random amount of tme, such a queung system s called retral queung system. Between trals the customer s called to be n orbt. Retral queung theory occupy a promnent role n the performance analyss of wde range of systems n telecommuncaton network, telephone swtchng and manufacturng. So, new type of retral queung models were proposed and solved by researchers n the feld of Mathematcs, Statstcs, Management and Engneerng. For a completed revew of man results and the lterature of mult server retral queues, one may refer the monograph by Faln and Templeton (997). The retral queues wth r servers and d watng postons are characterzed by when r servers are busy, an enterng customer occupes a watng poston wth total number of watng postons as d. If all the r+ d postons are occupedthe enterng customer leaves the system ot enters the orbt of nfnte capacty. The r server retral queue wth r+ d system sze Markovan s denoted by M / M / r/ r+ d. Cohen(957) studed extensvely n the M / M / r/ r retral queue. The statonary dstrbuton of mult server retral queues wth partcular values on r and d obtaned the closed form results. Jonn and Sedol (97) and Hanschke (987) derved n explct formula for the statonary probabltes of the M / M / / retral queue. A bvarate process was ntroduced by Gomez Corral and Ramalhoto (999), whch represents the ont process of number of servers and the watng poston occuped, and the number of customers n the orbt. They derved a closed form expresson for the statonary dstrbuton of the bvarate process usng a smple Correspondng author: M. Seenvasan, Assstant Professor, Annamala Unversty. Research felds: queung theory, stochastc process and ther applcatons. E-mal: emseen@redffmal.com.

2 64 The Performance Analyss Of A M/M//+ Retral Queue Wth Unrelable Server recursve approach. They also derve the statonary analyss of M/M//+ and M/M/3/3 queues wth lnear retral rates as partcular cases. The mult-server retral queue, based on the physcal behavor of the system, whch yelds an nfntesmal generator wth a modfed matrx geometrc equlbrum probablty vector by usng a smplfed approxmaton was analyzed by Neuts and Rao (99). They nvestgated a mult server retral queue by usng matrx geometrc method. The servers of a retral queung models, may subect to falures and repars. In queung system, t s normally assumed that the customer whose servce s nterrupted wll be able to complete hs servce when the nterrupton s cleared. In the retral queue, the behavor of the nterrupt customer s dfferent. That type of customer mght ether leaves the system or ons the orbt. Assan (988) and Kulkarn and Cho (99) ntroduced retral queues wth server falures and repars. The other notable works related to ths queueng model are Assan and Artaleo (998), Sherman and Kharoufeh (6) and Yang and L (994). Retral queung system wth unrelable server was analyzed by Kalyanaraman and Seenvasan (). The content of ths paper s the analyss of M / M / r/ r+ d, r retral queue wth unrelable server. For ths queue the Markov Process { t (): t } has been defned and the nfntesmal matrx Q has been obtaned. Usng the matrx equaton Q= and e=, the soluton vector s steady state, obtaned usng the Matrx Geometrc Method. The rest of the paper s organzed as follows: In secton, a two server retral queue, that s r=, has been analyzed. In secton 3, a performance measures are calculated n the above model. The model s analyzed usng numercal examples are gven n secton 4. The last secton contans a bref concluson. The Model and Analyss In ths secton, we consder two parallel servers are gvng servce to the customers wth one watng poston, the arrval follows Posson wth rate λ. On arrval f the arrvng customer fnd the system full, he leaves the servce areaand enters nto an orbt of nfnte capacty. From the orbt, the customer tres hs luck. α, beng the number of customers n the system, The nter-retral tme s negatve exponental wth rate {, }. The servce tme of each customer s agan negatve exponental wth rate ε. Whle dong servce the server may fal. The servers lfe tme s negatve exponental wth parameter β. On falure the server s sent for repar, the repar tme s negatve exponental wth parameter δ. Let () t = ( S() t, C() t, Q() t ) be the state of the process at tme t, Ct () s the number of customers both watng and beng served. Therefore, Ct ( ) {,,, 3}. Let Qt () be the number of customers n the orbt and Qt ( ) {,,, 3,...}. The elements of the state space of the process { () t = ( S() t, C() t, Q()) t : t } are St () =, the server s dle, St () =, the server s busy wth a customer, St () =, two servers busy servng a customer each, St () = 3, a server fals whle servng a customer, The state space of the process E s {{(,,) : } {(,,), (,,),(,3,) : } {(,,), (,3,) : } {(3,,), (3,,), (3,3,) : }} and retral rate α, (=,,). In equlbrum,the nfntesmal generator matrx s Q= ( q ab ) and s defned as

3 The Performance Analyss Of A M/M//+ Retral Queue Wth Unrelable Server 65 and the row addresses are defned usng Q B A A A A A A A =, a trdagonal matrx () = {(,, )(,,, )(,,, )(3,,, )(,,, )(3,,, )(3,,, )(3,,, )(33,,, )} : and agan s refned as = {(, )(,, )(,, )(3,, )(4,, )(5,, )(6,, )(7,, )(8,, )} The submatrces of Q are λ A = λ λ ( λ+ α) λ ε ( λ+ α + β + ε) λ β ( λ+ δ + β) λ δ β ( λ+ δ + β) δ β A = ε ( λ+ ε + β + α) λ β ε ( λ+ ε + β) β δ λ ( λ+ δ + α) δ ε ( λ+ δ + ε) λ δ ε ( λ+ δ + ε) α α A = α α and

4 66 The Performance Analyss Of A M/M//+ Retral Queue Wth Unrelable Server λ λ ε ( λ+ β + ε) λ β ( λ+ δ + β) λ δ β ( λ+ δ) δ B = ε ( λ+ β + ε) λ β ε ( λ+ β + δ) β δ λ ( λ+ δ) δ ε ( λ+ δ + ε) λ δ ε ( λ+ δ + ε) The steady state probablty vector for Q, f t exst, s ( ) =,,,, found by solvng Q = and e = () where e s the column unt vector of approprate dmensons. The system of lnear equatons Q = s solved usng the matrx geometrc method (Neuts, 978). Snce the rate matrx Q has a block trdagonal structure, each ( = 3,,,, ts) s a row vector of order nne, that s, =,,,,,,,,,. In the stable case, there exst a matrx R such that = R,. (3) Now the system Q = becomes B + A = and A + A+ A = + + ( ) B + RA = (4) R A + RA + R A =, (5) The vector s unquely determned by the equaton (4) and the normalzng equaton ( ) I R e= (6) The matrx R s the mnmal soluton to the matrx non-lnear equaton A + RA + R A =, (7) and t s an rreducble non-negatve matrx of spectral radus less than one. The followng teratve method can be used to compute R as follows.

5 The Performance Analyss Of A M/M//+ Retral Queue Wth Unrelable Server 67 R = (8) R = AA R AA, n (9) n+ n For a Markov process wth such generators, Neuts (978) has obtaned the stablty condton as πae< πae () where the row vector π = ( π, π, π, π3, π4, π5, π6, π7, π8) s obtaned from the nfntesmal generator A= A + A+ A ( λ+ α) ( λ+ α) ε ( λ+ β + α+ ε) ( λ+ α) β ( λ+ δ + β) λ δ β ( δ + β) δ β A = ε ( λ+ α + β + ε) λ+ α β ε ( β + ε) β δ α λ ( λ+ α+ δ) δ ε ( λ+ δ + ε) λ δ ε ( δ + ε) () It can be shown that A s rreducble and that the row vector π s unque such that πa= and πe= () Usng equaton () n equaton (), we have ( λ + α ) π + επ = (3) ( λ + α ) π ( λ + β + α ) π + επ + δπ = (4) 4 6 ( λ+ δ+ β) π + απ = (5) 6 λπ ( δ + β) π = (6) 3 ( λ + α ) π + δπ ( λ + α + ε + β ) π + επ + λπ + δπ = (7) δπ + ( λ + α ) π ( β + ε ) π + δπ = (8) βπ ( λ + α + δ ) π + επ = (9) 6 7

6 68 The Performance Analyss Of A M/M//+ Retral Queue Wth Unrelable Server βπ + βπ ( λ + δ ) π = () 4 7 βπ + βπ + λπ ( ε + δ ) π = () Solvng the above equatons, we get π + π+ π + π3+ π4 + π5 + π6 + π7 + π8 = () π = B δπ (3) (4) (5) 3 (6) 4 (7) 5 3 (8) 6 4 (9) 7 5 (3) 8 6 π δ δδ δ δ δ δ δ = ( ) (3) where B = ε /( λ+ α) δ = B6 δ = λ/ a3 δ = ab4 δ3 = ( βa + λb7 + B8ε) / β δ 4 = a δ 5 = B7 δ 6 = B8 a = ( λ+ β + α) a = ( λ+ δ + β) a = α a3 = ( β + δ) a4 = ( λ+ α + β + ε) a5 = ( λ+ α) a6 = ( β + ε) a7 = ( λ+ α+ δ) a8 = ( λ+ δ) a9 = ( ε + δ) a = a / a a λ a3 = / a8 = βε

7 The Performance Analyss Of A M/M//+ Retral Queue Wth Unrelable Server 69 a = βδ aa a = aε B = a8ε B = ( a9ε + aa 7 ) B3 = βa 9 7 B4 = ( B3δ Ba ) /( B3ε Ba ) B5 = ( aδ + Baa 4 5 ) B6 = ( aεb4 + aδ) / a B7 = ( βb6 + aa7 )/ ε and the stablty condton takes the form B = ( a β + B β + B )/ a λπ + α ( π + π ) < λ( π + π + π ) (3) Equaton (3) to (3) gves the steady state probabltes of A. Usng the Probablty Vectors calculated for the model M / M / / +. The performance measures are () The dle Probablty = Performance Measures ( {,,, 3,...}) the followng performance measures can be = () Pr{One Server s busy} = { } () Pr{Two Servers are busy} = { } 4 + = 5 = (v) Pr{ A Server s n repar condton} = { } = 8 (v) Blockng Probablty = { } = 8. (v) The Mean number of customers n the orbt = L = (v) The Varance number of customers n the orbt = V= (v) Mean Number of Customers n the orbt when the server s dle N = = e = e ( ) L = when one server s busy N = { } = when two servers are busy N = { 4 + 5} =

8 7 The Performance Analyss Of A M/M//+ Retral Queue Wth Unrelable Server when the server s n repar N 3 = { } λ s, = Numbercal Study Some numercal results related to the model dscussed n the above secton s presented here. By varyng α s, δ s, β s and ε s, three dfferent examples called Example, Example and Example 3 ofthe model, are gven below. Example : For λ =. 75, α =., α = 9., α = 6., α 3 =., ε = 5., β =., δ =., α = and the R matrx s gven by R = Usng ths R matrx, s calculated from the relaton ( B + RA ) = and the normalzaton condton ( I R) e =. The remanng vectors, =,,3...are obtaned from = R, = 3,,,... and are presented n the tables 5. and 5.. In the table rows,3,4,5 6,7,8,9, and represents the nne components of, =,,,... the last row represents the sum of the eght components. It s verfed that the total probablty s Table 5. Probablty Vectors Total Example : For λ =. 85, α =., α = 9., α = 6., α3 =., ε = 5., β =., δ =., α = and the R matrx s gven by 3 4 5

9 The Performance Analyss Of A M/M//+ Retral Queue Wth Unrelable Server R = Usng ths R matrx, s calculated from the relaton ( B ) + RA = and the normalzaton condton ( ) I R e =. The remanng vectors, =,,3...are obtaned from = R, = 3,,,... and are presented n the tables 5.3 and 5.4. In the table rows,3,4,5 6,7,8,9, and represents the nne components of verfed that the total probablty s , =,,,... the last row represents the sum of the eght components. It s Table 5. Probablty Vectors Total Table 5.3 Probablty Vectors Total

10 7 The Performance Analyss Of A M/M//+ Retral Queue Wth Unrelable Server Table 5.4 Probablty Vectors Total Example 3: For λ =. 95, α =., α = 9., α = 6., α 3 =., ε = 5., β =., δ =., α = and the R matrx s gven by R = Usng ths R matrx, s calculated from the relaton ( B ) + RA = and the normalzaton condton ( ) I R e =. The remanng vectors, =,,3...are obtaned from = R, = 3,,,... and are presented n the tables 5.5, and 5.6. In the table rows,3,4,5 6,7,8,9, and represent the nne components of verfed that the totalprobablty s , =,,,... the last row represents the sum of the eght components. It s The performance measures for the numercal models related to examples,,3 are calculated, for the model M / M / / + the dle probabltes, the blockng probabltes, probablty that one server are busy, the total retral rate, and the mean number of customers n the orbt are gven n the table 5.7. The mean number of customers n orbt s also too hgh n the case of two server queues. The results show that the number of servers make substantal changes n the performance measures.

11 The Performance Analyss Of A M/M//+ Retral Queue Wth Unrelable Server 73 Table 5.5 Probablty Vectors Total Table 5.6 Probablty Vectors Total Table 5.7 Performance Measures No. Measures for Model Example Example Example3. The dle Probablty Pr {One server s busy} Pr {Two servers are busy} Blockng Probablty Pr {A server s n repar condton} The Mean number of Customers n the orbt (L) The Varance number of Customers n the orbt The Mean number of Customers n the orbt when the server s dle ( N ) when one server s busy ( N ) when two servers are busy ( N ) when three servers are busy ( N 3)

12 74 The Performance Analyss Of A M/M//+ Retral Queue Wth Unrelable Server Concluson In ths paper we have consdered a mult server retral queung system. We have obtaned the steady state probablty vector by applyng matrx geometrc method. Usng the steady state probablty vector, the mean number of customers n the orbthas been obtaned. Furthermore, we have performed numercal analyss by assumng partcular values to the parameter. References Assan.A On the M/G// queueng system wth repeated orders and unrelable server, Journal of Technology 6, 98-3, 988 Assan.A and J.R. Artaleo, On the sgle server retral queue subect to breakdowns, Queueng Systems 3, 37-3, 998 Cohen, J.W. Basc problems of telephone traffc theory and the nfluence of repeated calls, phllps Telecommuncaton Revew 8, 49-, 957. Faln, G.I. and J.G.C. Templeton, Retral Queues, Chapman and Hall, 997. Gomez-Corral, A. and M.F.Ramalhoto, The statonary dstrbuton of a Markovan process arsng n the theory of mult server retral queung systems, Mathematcal and Computer Modelng 3, 4-58,999. Hanschke, T. Explct formulas for the characterstcs of the M / M / / queue wth repeated attempts, Journal of Appled Probablty 4, , 987. Jonn, G.L. and J.J. Sedol, Telephone systems wth repeated calls, In proceedngs of the 6-th Internatonal Teletraffc Congress, Munch, 435(-5), 97. Kalyanaraman, R. and M.Seenvasan, Mult-server retral queung system wth unrelable server, Internatonal Journal of computatonal cognton Vol 7, No 3, 3-,. Kulkarn, V.G. and B.D. Cho, Retral queues wth server subect to breakdowns, Queueng Systems 7, 9-8, 99. Neuts, M.F., Markov chans wth applcatons n queung theory,whch have Matrx- Geometrc nvarant probablty vector,adv.appl.probab., 85-, 978. Neuts, M.F. and B.M. Rao, Numercal nvestgaton of a mult server retral model, Queung Systems 7, 67-89, 99. Sherman, N.P. and J.P. Kharoufeh, An M/M/ retral queue wth unrelable server, Operatons Research Letters 34,697-75, 6. Yang, T. and H.L., The M/G/ retral queue wth the server subect to startng falures, Queueng Systems 6, 83-96, 994.