Steady-state Behavior of a Multi-phase M/M/1 Queue in Random Evolution subject to Catastrophe failure

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1 Advaces Theoretcal ad Appled Mathematcs ISSN Volume, Number 3 (26), pp Research Ida Publcatos Steady-state Behavor of a Mult-phase M/M/ Queue Radom Evoluto subect to Catastrophe falure M. Re Sagayara, S. Aad Gaa Selvam, Mogara. D R. Reyald Susaatha 2 Departmet of Mathematcs, Sacred Heart College(Autoomous), Trupattur-6356, Vellore Dstrct, Taml Nadu, S. Ida. 2 Recktt Beckser, Ida Lmted (Gurgao), New Delh. Ida. Emal: re. sagaya@gmal. com Abstract I ths paper, we cosder stochastc queueg models for Steady-state behavor of a mult-phase M/M/ queue radom evoluto subect to catastrophe falure. The arrval flow of customers s descrbed by a marked Markova arrval process. The servce tmes of dfferet type customers have a phase-type dstrbuto wth dfferet parameters. To facltate the vestgato of the system we use a geeralzed phase-type servce tme dstrbuto. Ths model cotas a repar state, whe a catastrophe occurs the system s trasferred to the falure state. The paper focuses o the steady-state equato, ad observes that, the steady-state behavor of the uderlyg queueg model alog wth the average queue sze s aalyzed. AMS Subect Classfcato: 58Kxx, 6J2, 74Gxx Keywords: M/M/ Queue, M/G/, Mult-phase, Radom evoluto, Steadystate equato, Catastrophe falure. Itroducto I real lfe, may queug stuatos arse whch are ot relable ad Catastrophe may occur leadg to loss of several or all customers. Such stuatos are commo computer etwork applcatos, telecommucato applcatos that deped o satelltes ad vetory system that store pershable goods. Customer mpatece s also observed queue models lke mpatet telephoe swtch board customers, packet trasmsso etc Also may tradtoal studes aalyze queug system steady state, requrg approprate warm up perod.

2 24 M. Re Sagayara et al However, may cases the system beg modeled ever reaches steady state ad hece do ot accurately portray the system behavor, as mltary ar traffc cotrol, emergecy medcal servce etc The earler works o the traset behavor of queues lterature were publshed the late 95 s ad early 96 s. The traset soluto of varous sgle server queue models lke state depedet queues [4], potetal customers dscouraged by queue legth[3], feedback wth catastrophes [8] etc are studed the lterature. I [] derved traset soluto of a sgle server queue wth system catastrophe ad customer mpatece [6]. A queueg system wth ths costrat s of much terest ad s studed by may authors. Such models are useful for the performace evaluato of commucatos ad computer etworks whch are characterzed by tme-varyg arrval, servce ad falure rates. To corporate the effects of exteral evrometal factors to a stochastc model a queue wth a radom evromet s cosdered. The radom evromet process may take a umber of vared forms such as a dscrete-or cotuous-tme Markov cha, a radom walk, a sem-markov process, or a Browa moto, etc. If the radom evromet s Markova, the prmary stochastc process to whch t s attached s sad to be Markov-modulated. [5] cosdered a fte-capacty storage model wth two-state radom evromet ad characterzed ts steady-state behavor. [4] Formulated a queueg system uder batch servce a M/G/ System ad assumed that the batch sze depeds o the state of the system ad also o the state of some radom evromet whereas [] cosdered a queueg system a sem-markova radom evromet. [7] Cosdered a M/M/ queueg system wth catastrophes ad foud the traset soluto of the system cocered. [8] aalyzed dscouraged arrvals queue wth catastrophes ad derved explct expressos of the traset soluto alog wth the momets. [] Studed a queue wth system Catastrophes ad customer mpatece ad derved ts system-sze probabltes usg cotued fracto techology. Whe occurs, a falure causes all preset obs to be cleared out of the system ad lost. The system tself the moves to a repar phase that ts durato s expoetally dstrbuted. Beg repared, the system moves to a operatve phase wth probablty q where q =. Istead of havg oe repar state f we have a checkg state to check the overall codto of the system after gettg repared the effcecy of the model wll get mproved. Hece our paper we have cosdered a M/M/ mult-phase queueg model radom evoluto wth Catastrophe. Ths model cotas + operatoal uts cludg a repar state followed by a checkg state. Wheever a Catastrophe occurs the system moves to the repar state ad after gettg repared t moves to the checkg state wth probablty oe. From the checkg state t moves to ay oe of the remag uts. For the above model steady-state behavor s aalyzed. The rest of ths paper s orgazed as follows. I secto 2, we descrbe the mathematcal model. Secto 3 deals wth the steady-state equato of the model uder cosderato are aalyzed. The above probabltes are aalyzed usg geeratg fuctos Secto 4. I secto 6, we aalyze the performace measures

3 Steady-state Behavor of a Mult-phase M/M/ Queue 25 ad cases are dscussed. Fally, some cocludg remarks. 2. Model Descrpto I ths paper, we cosder a M/M/ queueg model wth + uts operatg a radom evromet. These + operatg uts form a cotuous tme Markov cha wth uts =,, 2, 3,,,, ad the correspodg Trasto Probablty Matrx as gve below.... q2 q3... q The durato of tme the markov cha stays phase s a expoetally dstrbuted radom varable wth mea =. Whe the system eds ts soour perod phase, t umps to phase wth probablty q. Occasoally a falure occurs whe the system s ut 2. At that stat the system s trasferred to the ut = ad the to =. The tme spet by a system at the falure ut = s a expoetally dstrbuted radom varable wth mea η. The tme take for the verfcato process at = s also a expoetally dstrbuted radom varable wth mea η After the. completo of the verfcato process the system s trasferred to ay oe of the operatg uts 2 wth probablty q ad hece 2 q =. The arrval rate for each of the falure uts = ad = are λ ad λ respectvely. As there s o servce these uts μ = μ =. I each actve ut = 2 to, the system stays utl a catastrophe occurs whch seds t to ut ad the to ut. The above system ca be represeted by the stochastc process {U(t), N(t)} where U(t) deotes the ut whch the system operates at tme t ad N(t) deotes the umber of customers preset at tme t. The system s sad to be state (, m) f t s phase, ad there are m customers the system. The steady state probabltes of the system beg state wth m customers s deoted by p. That s m { } p = lm P( U( t) =, N( t) = m) t,, m=,, 2... (2.) m t 3. Steady State Equatos: The system s steady-state balace equatos are gve as follows: For the falure phase =, whle m =,

4 26 M. Re Sagayara et al ( ) λ + η p = η p = η p m. (3.) Ad whle m ( λ + η) p, (3.2) m = λp m For =, 2.,, ad m =, ( λ + η) p = μp. + ηqp (3.3) Ad whe m, ( λ + μ + η) pm = λp, m + μp, m+ + ηqp m (3.4) From () ad (2) we get that m λ pm = pm (3.5) λ + η λ p = p (3.6) λ + η Where p = p,,,2,3...,. m The lmt probabltes of the uderlyg MCQ, m= { } d = lm P( U( t) = ) satsfy t d = d = = d ad d = dq for. Therefore q d = ad d = for for. Hece the proporto of tme the system resdes 2 2 phases gve by d q η η p = = (3.7) dk qk + k= ηk η k= ηk d η η p = = = (3.8) dk qk αη + k= ηk η k= ηk qk Where α = η + k= η k From (7) t follows that ηp = η qp,, 2,3...,. Now gve p p s calculated from (6), It s p = (3.9) α( λ + η) Ad all p m for m are explctly determed by eq. (5). The system s postve recurret because of catastrophe effect. =

5 Steady-state Behavor of a Mult-phase M/M/ Queue Geeratg fuctos: m= Defe m G = p z, =,..., z (4.) m Settg = (4. ), we get λ + γ G = p (4.2) λ( z) + γ usg (3. ), (4. 2) modfes to λ γ G γ p (4.3) + = 2 m Settg z = ad = (4. ) ad usg (4. 3) we get γ pm = γ pm (4.4) m= 2 For = the equatos correspodg to (4. 2), (4. 3) ad (4. 4) are obtaed as λ+ γ G = p (4.5) λ( z) + γ ( λ γ ) G γ p (4.6) + = 2 m= γ p = γ p m m m m= 2 Equato (4. 4) ad (4. 7) cocdes mplygγ p m = γ p (4.7) m. Usg equato (3. m= m= 5) we get G[ λz( z) + μ( z ) + zγ] γqg = pμ( z ), 2 (4.8) Now, G ( z ) ad G ( ) z ca be determed from (4. 3) ad (4. 6), each G 2 ca be foud from (4. 8) f p s kow. Defe f = λ ( z) + γ f = λ( z) + γ f = λz( z) + μ( z ) + γ, 2 The quadratc polyomal f ( z ), I 2 each have two roots. Let z deote oly postve roots of f the terval (, ). The we have 2 ( λ + μ + γ) ( λ + μ + γ) 4λμ z = (4.9) 2λ I fact, z represets the Laplace-Steltes Trasform, evaluated at a pot γ of the busy perod a M/M/ queue wth arrval rate λ ad servce rate μ Substtuto of (4. 9) (4. 8) gves

6 28 M. Re Sagayara et al p γqzg = μ ( z) Usg (4. 6) modfes to 2 p mqz m= γ p =,2 ( λ( z) + γ) μ( z) Now, each G ( z ) ca be completely determed usg p o 5. Mea queue sze: Let ' m m= G () = E[ L] = mp, =,... (5.) λ λ λ EL [ ] = = + p (5.2) q k 2 η η η + η k= ηk ( λ + μ + η) p + ηe[ L] η q( p + E[ L ]) = μ p, (5.3) Ths leads to λ λ μ μ EL [ ] = [ μp + + ] (5.4) η q η η η k + η η k= k λ λ [ μ = μp + + pη ] (5.5) η η η η The total umber of customers the system s = EL [ ] EL [ ] 2 λ λ η p qz λ λ μ = + p p η (5.6) η η η ( λ( z) + η)( z) η η η Performace measure: Let M be the umber of customers cleared from the system per ut tme. The EM [ ] = η mp = η EL [ ] m m= The fracto of customers recevg full servce s therefore λ EM [ ] EM [ ] = λ λ

7 Steady-state Behavor of a Mult-phase M/M/ Queue Case Whe the arrval process stops wheever the system s dow, that s, λ =, mplyg that p,. m= m. Usg the steady-state equatos we get for = ηp = ηp (6.) Remas uchaged, but for ad m s replaced by ( λ + μ + η) pm = λp, m + μp, m+ (6.2) The set of probabltes p s gve by ηg = ηp (6.4) Ad for Defe f = η ( λ ( zz ) + μ ( z ) + η zg ) = μ ( z ) p + qη p z (6.5) f = ( λ z μ )( z) + η z =,..., Where A(z) s gve by f... f... f2... Az ( ) = f ( ) z Ad the vectors g(z) ad b(z) are gve by G ηp G μ p( z ) + ηq pz.. gz ( ) = bz ( ) =.... G μp( z ) + ηqpz The the equvalet s G = p G = [ μ ( z ) p + η q p z f Dfferetatg

8 22 M. Re Sagayara et al EL [ ] = ηe[ L] = μp + ηqpz+ ( λ μ η) p Sce ths case we have thatηp = η qp. Substtutg ths relato xxxx yelds ηel [ ] + μ( p p) = λp Whch aga equates the flow ad outflow rates of phase-i, oly tme t does ot cocerg the phase trasto testy. Cocluso: I ths paper, we have troduced stochastc queueg models for Steady-state behavor of a mult-phase m/m/ queue radom evoluto subect to catastrophe falure. We are curretly vestgatg the use of more geeral servce tme dstrbutos. Mea queue szes, mea watg tmes were calculated. However, we are vestgatg varous approaches for valdato. Referece: [] E. Altma ad U. Yechal, Aalyss of customer s mpatece queues wth server vacatos, Queueg Systems, 52, No. 4 (26), [2] Baykel-Gursoy M. ad Xao W. Stochastc Decomposto M/M/ queues wth Markov Modulated Servce Rates, Queueg Systems, 48(24), [3] X. Chao, A queueg etwork model wth catastrophes ad product form soluto, Operatos Research Letters, 8, No. 2 (995), [4] R. V. Kakubava, Aalyss of queues uder batch servce a M/G/ system a radom evromet, Automato ad Remote Cotrol, 62, No. 5 (2), [5] B. Krsha Kumar, ad D. Arvudaamb, Traset soluto of a M/M/ queue wth catastrophes, Computers ad Mathematcs wth Applcatos, 4, No. (2), [6] R. Sudhesh, Traset aalyss of a queue wth system dsasters ad customer mpatece, Queueg Systems, 66, No. (2), [7] X. W. Y, J. D Km, D. W. Cho, K. C. Chae, The Geo/G/ queue wth dsasters ad multple workg vacatos, Stochastc Models, 23, No. 4 (27), [8] Parthasarathy P. R., Sudhesh. R, Exact traset soluto of state depedet brth death processes, J. appl. Math. Stch. Aal. 26, -6. [9] Parthasarathy P. R., Sudhesh. R, Traset soluto of a mult-server Posso queue wth N-Polcy, Computers ad Mathematcs wth Applcatos, 55(28), [] Rau S. N., U. N. Bhat, A computatoally oreted aalyss of the G/M/ queue, Operato research, 9(982), [] Sudhesh. R, Traset aalyss of a queue wth system dsasters ad customer mpatece, Queueg System, 66(2), 95-5.

9 Steady-state Behavor of a Mult-phase M/M/ Queue 22 [2] Thagara. V. Vatha. S, O the aalyss of M/M/ feedback queue wth catastrophes usg cotued fractos, Iteratoal oural of Pure ad Appled Mathematcs 53(29), [3] Ur Yechal, Queues wth system dsasters ad mpatet customers whe the system s dow, Queueg System, 56(27), [4] T. TakeSgle-server queues wth Markov-modulated arrvals ad servce speed. Queueg Systems, 49 (25), [5] T. Take ad T. Hasegawa:A geeralzato of the decomposto property the M/G/ queue wth server vacatos. Operatos Research Letters, 2 (992), [6] R. W. Wolff: Posso arrvals see tme averages. Operatos Research, 3 (982), [7] D.-A. Wu ad H. Takag: M/G/ queue wth multple workg vacatos. Performace Evaluato, 63 (26), [8] W. S. Yag, J. D. Km, ad K. C. Chae: Aalyss of M/G/ stochastc clearg systems. Stochastc Aalyss ad Applcatos, 2 (22), 83-.

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