Steady-state Analysis of the GI/M/1 Queue with Multiple Vacations and Set-up Time
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1 Moder Applied Sciece September 8 Steady-state Aalysis of the GI/M/ Queue with Multiple Vacatios ad Set-up Time Guohui Zhao College of Sciece Yasha Uiersity Qihuagdao 664 Chia zhaoguohui8@6com Xixi Du College of Sciece Yasha Uiersity Qihuagdao 664 Chia Naishuo Tia College of Sciece Yasha Uiersity Qihuagdao 664 Chia Xiaohua Zhao College of Sciece Yasha Uiersity Qihuagdao 664 Chia Dogmei Zhao College of Sciece Yasha Uiersity Qihuagdao 664 Chia Abstract I this paper we cosider a GI/M/ queueig model with multiple acatios ad set-up time We derie the distributio ad the stochastic decompositio of the steady-state queue legth meawhile we get the waitig time distributios Keywords: Multiple acatios Set-up time Stochastic decompositio Itroductio Vacatio queues serers to stop the customers serice at some periods ad the time durig which the serice is iterrupted is called the acatio time Vacatio queue research origiated from Ley ad Yechial the may researchers o queuig theory deal with this fields So far the theory frame whose core is the stochastic decompositio is deeloped ad acatio queues hae bee applied successfully to may fields such as computer systems commuicatio etworig electroic ad call ceters Details ca be see i the sureys of Doshi ad the moographs of Tia For GI/M/ type queues with serer acatios Tia used the matrix geometric solutio method to aalyze ad obtaied the expressios of the rate matrix ad proed the stochastic decompositio properties for queue legth ad waitig time i a GI/M/ acatio model with multiple expoetial acatios Descriptio of the model Cosider a classical GI/M/ queue iter-arrial times are iidrs Let A (x ad a ( s be the distributio fuctio ad LS trasform of the iter-arrial time A of customers The mea iter-arrial time is E( A a ( λ Serice times durig serice period acatio times ad set-up times are assumed to be expoetially distributed with rate µ θ β respectiely We assume that the serice disciplie is FCFS Suppose τ be the arrial epoch of th customers with τ Let L ( L τ be the umber of the customers before the th arrial Defie 57
2 Vol No 5 Moder Applied Sciece The process {( L J } We itroduce the expressios below the th arrial occurs durig a serice period J J ( τ the th arrial occurs durig a set - up period the th arrial occurs durig a acatio period is a Maro chai with the state space { U j j } Ω ( ( a µ µ t β x [ ] µ ( tx da( t β ( t e! t µ ( t x b e e dxda( t! t t y [ µ ( tx y ] ( θx β y µ txy c θe βe e dxdyda( t! First the trasitio from( i to ( j occur if i + j serices complete durig a iter-arrial time Therefore we hae Similarly p(( b i j L i+ i j i+j p( ( a + i j i + i j i j L θt p(( i i+ e da( t a ( θ i j i+j t θx β( tx (( i i+ θ θ β θ ( θ β α p( ( c i j L i+ p e e da( t ( a ( a ( ( The trasitio matrix of ( L J ca be writte as the Bloc-Jacobi matrix B A B A A P B A A A B A A A A M M M M M O where B c α( A ( c α( p βt ( i( i+ e da t a ( β ( a A b c α( a A b c ai i B bi i ci α ( i The matrix P is a GI/M/ type matrix Steady-state queue legth distributio Lemma If ρ λµ < θ β > the δ > θβ ( δ > θ β where a ( β δ β µ ( a ( β β µ ( a ( θ ( θ µ [ a ( θ] θ µ ( Theorem If ρ < θ β > the the matrix equatio R R A has the miimal o-egatie solutio 58
3 Moder Applied Sciece September 8 R βδ αβ( δ α ( where is the uique roots i the rage < z < of the equatio z a ( µ ( z α θ ( θ β δ ad are defied as i ( Proof Because all A are lower triagular we assume that R has the same structure as r R r r r r r we obtai r r a a r r r r r r r a r b r r r r r a + r r r r r a ( µ ( i i + α ( i i i i i j i j i i j i i + r rr b + rc i ( As we ow if ρ < θ > the first equatio has the uique root r i the rage < r < We ca compute β [ a ( µ ( r r ] β [ a ( µ ( ] rb i i a ( µ ( rr a a β µ ( r β µ ( i a ( µ ( Fially we obtai r βδ r αβ( δ ad the expressio for R Theorem The Maro chai ( L J is positie recurret if ad oly if ρ < θ β > Proof Based o Neuts the Maro chai ( L J is positie recurret if ad oly if the spectral radius SP(Rmax{ } of R is less tha ad the matrix B A + + B[ R] R B R A has a positie left iariat ector Eidetly SP(Rmax{ } < Substitutig the expressios for R A ad B i B[R]we obtai c c a a BR [ ] βδa b βδ a b + A B α Where δ δ ( c A αβ + a α + α 59
4 Vol No 5 Moder Applied Sciece ( δ δ c B αβ + a + α It ca be erify that B[R]has the left iariat ector K( αβ( δ α ( ( Thus if ρ < θ β > the Maro chai ( L J is positie recurret If ρ < θ β > let ( L J be the statioary limit of the process ( L J Let ; Theorem If < j ( P{ L J j} lim P{ L J j} ( j Ω ρ the statioary probability distributio of ( J L is Kαβ δ Kα( K where ( ( ( K αβ ( δ ( + α( ( + ( ( Proof is gie by the positie left iariat ector ( ad satisfies the ormalizig coditio The we get ( + ( ( I R e ( ( ( ( ( + ( ( + ( ( K δ α We obtai We hae Fially we obtai the theorem K ( K αβ δ α ( ( ( ( ( R Theorem 4 If ρ < the statioary queue legth L ca be decomposed ito the sum of two idepedet radom ariables: L L + L where L is the statioary queue legth of a classical GI/M/queue without acatio follows a d geometric distributio with parameter ; L follows the discrete P distributios d ( ϕ T of order where K + βδ ( α ( + ϕ α αβ K ϕ T T Proof The PGF of L is as follows: L ( z z P( L 6
5 Moder Applied Sciece September 8 α ( ( ( z βδ β K + z + z z α K z + zβδ ( α ( z + zαβ α L( z Ld ( z z z z Where L(z is the PGF of L of a classical GI/M/queue without acatio L d (z K z + zβδ ( α ( z + zαβ α z z z + zβδ ( z + zβδ z + ( + βδ z z (4 Substitutig the aboe equatio ito (4we obtai the distributio of We ca get meas Waitig time distributio L d K + βδ ( α ( + αβ EL ( + α ( ( Let W ad W ( s be the steady-state waitig time ad its LST respectiely Firstly let be the probability that the serer is i the serice(set-up acatio period whe a ew customer arries We ca compute ( δ ( ( ( + ( ( + ( ( δ α α( ( ( ( + ( ( + ( ( ( ( ( ( + ( ( + ( ( δ α δ α Theorem 5 If ρ < θ β > the LST of statioary waitig time W is W ( s β ( µ + s( µ ( + θ µ ( β β + s µ ( + s µ ( + s θ + s µ ( + s β + s ( δ s ( µ + s( ( µ ( + s + s + s + µ + δ ( ( µ ( µ ( µ ( Proof Whe a customer arries if there are customers ad the serer is i the serice period the waitig time equals serice times by the rate µ The we hae µ W ( s Kαβ δ µ + s ( δ s ( µ + s( ( µ ( µ + (5 δ ( ( µ ( + s µ ( + s µ ( + s Whe a customer arries if there are customers ad the serer is i the set-up period the waitig time is the sum of the residual set-up time ad serice times by the rate µ The we hae β µ W( s Kα ( β + s µ + s 6
6 Vol No 5 Similarly Moder Applied Sciece β ( µ + s( µ ( (6 β + s µ ( + s µ ( + s W( s K From (5-(7 we hae the result i Theorem 4 With the structure i Theorem 4 we ca get the expected waitig time E (W θ µ ( β µ ( µ ( β + 4 Numerical examples % µ β θ θ µ ( β (7 µ + s β + sθ + s θ + s µ ( + s β + s δ + + µ ( µ [ δ( ( ] µ ( µ ( I the aboe aalysiswe obtai the expected queue legth i the steady state The differece of parameters may ifluece the queue legth So we preset umerical examples to explai Refereces Doshi BT(986Queuig systems with acatios-a surey Queuig Syst 9-66 Neuts M (98 Matrix-Geometric Solutios i Stochastic models Johs opis Uiersity Press Baltimore Qi X (6 GI/Geo/ discrete-time queue with set-up period ad multiple acatios Operatio Research Tia NS (6Vacatio Queueig Models: Theory ad Applicatios Spriger-Verlag New Yor Tia NS (989 The GI/M/queue with expoetial acatios Queueig Syst 5-44 Tia NS( The discrete time GI/Geo/ queue with multiple acatios Queueig Syst Figure The relatio of E(L with β 6
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