On Poisson Bulk Arrival Queue: M / M /2/ N with. Balking, Reneging and Heterogeneous servers

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1 Applied Mathematical Scieces, Vol., 008, o. 4, O oisso Bulk Arrival Queue: M / M // with Balkig, Reegig ad Heterogeeous servers M. S. El-aoumy Statistics Departmet, Faculty of Commerce, Dkhlia, Egypt Al-Azhar Uiversity, Girls Brach. drmahdy_elpaoumy@yahoo.com Abstract The aim of this paper is to derive the aalytical solutio of the queue: M / M // for batch arrival system with balkig, reegig ad two heterogeeous servers. A modified queue disciplie to the classical oe FIFO is used with a more geeral coditio. The steady-state probabilities ad some measures of effectiveess are derived i explicit forms. Also some special cases are deduced. Keywords: Heterogeeous servers, batch arrival, balkig, reegig Itroductio Abou-El-Ata ad Al-Seedy [] studied the system: M / M / with both balkig, ad reegig cocepts, Cromie et al [3] discussed the queue: M / M / Cwithout ay cocepts ad Al-Seedy [] are treated the system M / M // with both balkig ad a additioal server for loger queue. The preset paper treats the aalytical solutio of the queue: M / M / / with batch arrival queues cosiderig balkig, reegig, heterogeeity ad differet probability i choosig the servers.

2 70 M. S. El-aoumy Aalyzig the model I this work, it is assumed that the uits arrive at system i batches of radom size, i.e., at each momet of arrivals there is a probability cj = p( x= j) that j uits arrive simultaeously ( cj = ), c 0 = 0 ad the iterarrival times of batches follow a j = egative expoetial distributio with time idepedet parameter λ. Let λ cj Δ t, ( j =,..., ) be the first order probability that a batch of j uits comes i time Δ t. We assumed that we have a fiite strog room such that the total umber of customer i the system is o more tha ad two heterogeeous servers differet rates μ, ad μ. The queue disciplie cosidered here is modificatio of both Sigh [5] ad Krishamoorhi [4], ad it is: i) If both servers are free, the head customer of the queue goes to the first sever with probability π or to the secod sever with probability π, π + π =. ii) If oly oe server is free, the head uit goes to directly to it. iii) If the two servers are busy, the uits i their order util ay server become vacat. Cosider the balk cocept with probability β = prob. {a uit jois the queue}, where 0 β < if =,3,..., ad β = if = 0,. We assume that the uit may reege accordig to a expoetial distributio, t f () t = α e α, t > 0, with parameterα.the probability of reegig i a short period of time Δ t is give by r = ( ) α Δt, for =,3,..., ad r = 0, for = 0,,. 3 The steady state equatio ad their solutio We defie the equilibrium probabilities: 0,0,0 0, = prob. {there is o uit i the system}, = prob. {there is o uit i the system}, = prob. {there is o uit i the system}, = prob. {there is o uit i the system}, =,3,..., Also, 0 = 0,0, =,0 + 0, ad =,.

3 oisso bulk arrival queue 7 Usig the rate out = rate i approach, the steady state differece equatio ca be writte as follows: λ0 = μ.0 + μ0., = 0 () ( ) ( ) λ μ μ λcπ λ μ μ λ π +.0 = =. + c 0 ( ) , = βλ + μ = ( μ + α) + λc+ λc, = (3) ( ) ( ) βλ + μ + α = μ + α + βλ c + λc + λc, = 3, 4,..., (4) j j 0 j= ( ) μ + α = βλ c + βλ c+ λc + βλc + λc + λc, = 3 j j j i j 0 j= j= i= j+ j= where μ3 = μ+ μ. From equatios () ad () we have: λ[ λ+ μ( c) + cπμ3],0 = 0 μ λ+ μ [ 3] [ + ( c) + c 3] μ [ λ+ μ ] λ λ μ π μ = 0, 0 3 Therefore, () (5) (6). (7), =Δ 0 (8) where [ λμ + ( c )( μ + μ ) + c μ ( π μ + π μ )] λ 3 3 Δ =. μμ [ λ + μ3 ] Summig () ad () we get: λ = [ + ( c) 0] (9) μ3 We ca writte the steady-state equatios as follows = Δ, = 0 [ + ( c ) ] = ϕ, = 0 = [ + θ ] βϕ c ϕ ( c + c ), = 3, 4,..., (0) S S 0 S= where βλ θ = α, =,,...,, = λ, μ3 + α μ3 + α = 0,,... ut = g0 i equatio (0) we deduce that: ( ) ϕ ( )

4 7 M. S. El-aoumy = 0 Δ = g = ϕ0 Δ+ ( c) = ( + θ ) g βϕ c sgs ϕ ( c g + c g0) = 3,4,..., s= = 0 From the boudary coditio: 0 = + g. = =, we get Thus, the expected umbers of uits i the system ad i the queue are, respectively., L = g, L = L q 0 = ad the expected waitig time i the system ad i the queue, respectively, L L q W =, W q =, λ λ where μ λ = 3 ( L L ), q μ3 = μ+ μ. 4 Special cases Some queuig systems ca be obtai as special cases of this model. i) If μ = μ, π = π =/ ad c j = δ j where δ j is the Kroecker delta fuctio we get the homogeeous servers queue: M / M / / with balkig ad reegig. Moreover if α = 0 ad β =, we have the queue M / M / / without balkig ad reegig which studied by white et al.[8], Medhi [6] ad Buday [3]. ii) If we put c j = δ jwe obtai the heterogeeous servers queue: M / M / with balkig ad reegig, Moreover, if, i.e. i the ifiite capacity space case, β = ad α = 0 we have the queue M / M / without balkig ad reegig but with heterogeeous servers, which studied by Krishaoorthi [5]. While, if, α = 0 π = ad π = 0 we get the heterogeeous servers queue : M / M / with balkig oly, which discussed by sigh [7].

5 oisso bulk arrival queue 73 5 umerical Example I the above system, lettig = 5, i.e., the queue: M / M / / 5 the results are: ϕ ( c ) { ( θ ) ( θ )( θ ) ( θ )( θ )( θ ) ( ) ( ) } [ ] ( ) ( )( ) ϕ[ cδ+ c3]. { + ( + θ3) } ϕ3[ c3δ+ c4], 0{ ϕ0 ( ) 3( θ) 4( θ)( θ) 4βϕ 5( θ)( θ)( θ3) ( ) ( ) ] [ ] ( ) ( )( ) ϕ[ cδ+ c3] 4+ 5( + θ3) 5 ϕ3[ c3δ+ c4]}, = +Δ+ Δ { } βϕ c + θ βϕ c + θ βϕ c βϕ c ϕ cδ+ c + + θ + + θ + θ βϕ c L= Δ+ Δ+ c c βϕc + θ3 5βϕ3c + θ 5βϕ3c ϕ cδ+ c θ θ + θ3 5βϕ3c Lq = L + +Δ, 0 0 ad the expected waitig time i the system ad i the queue are, respectively, L Lq W =, Wq = μ3 μ3 ( 0 Δ0 ) ( 0 Δ0 ) ow, we itroduce the three tables for some measures of effectiveess at μ = 8, μ = 0, c = 0.3, c = 0., c3 = 0., c4 = 0.8, c5 = 0.6, ad λ = for the differet values of π, β adα whe two of them are fixed. Table I α = 0.05, β = 0. C = 0.3, C =0., C 3 =0., C 4 =0.8 ad C 5 =0.6. Π 0 L Lq W Wq

6 74 M. S. El-aoumy Table II α = 0.05, π = 0. C = 0.3, C =0., C 3 =0., C 4 =0.8 ad C 5 =0.6. Β o L L q W W q Table III π = 0., β = 0. C = 0.3, C =0., C 3 =0., C 4 =0.8 ad C 5 =0.6. α o L L q W W q

7 oisso bulk arrival queue 75 6 Coclusio I this paper, the batch arrival model: M / M / / is studied with balkig, reegig ad heterogeeous servers. The recurrece relatio for g that gives all the probabilities iterms of 0 which ca be determied from the boudary coditio. We discussed the example ad deduced the expected umber of uites i the system, i the queue, waitig time i the system ad i the queue. Refereces [] M.O. Abou-El-Ata ad R.O. Al-Seedy, "The Bulk Arrival Queue: with Reegig ad Balkig" Joural of the faculty of Educatio, o. 4, 989. [] R.O.Al-Seedy, "The Trucated Queue: M/M//K with Both Balkig ad A Additioal Server" Mkrolectro. Reliab., Vol. 3, o. 6, (99)., [3] B.D. Buday, "A Itroductio to queuig theory" Joh Wiley, ew York, 996. [4] M.V. Cromie, M.L. Choudhry ad W.K. Grassma "Further Results for the Queuig System " J. oper. Res. Soc., Vol. 30, o. 8, (979) [5] B. Krishamoorthi, "O oisso queue with two heterogeeous servers". Oper. Res., o., (963) ,. [6] J. Modhi, "Stochastic models i queuig theory" Academic press, ew York, 99. [7] V.. Sigh, "Two-Server Markovia queues with balkig: heterogeeous VS. homogeeous servers", oper, Res., o. 8,( 970) [8] J.A. White J.W. Schmidt ad G.K. Beett, "Aalysis of queuig systems", Academic ress, ew York, 975. Received: May 6, 007

B. Maddah ENMG 622 ENMG /27/07

B. Maddah ENMG 622 ENMG /27/07 B. Maddah ENMG 622 ENMG 5 3/27/7 Queueig Theory () What is a queueig system? A queueig system cosists of servers (resources) that provide service to customers (etities). A Customer requestig service will