On a Truncated Erlang Queuing System. with Bulk Arrivals, Balking and Reneging

Transcription

1 Appled Mathematcal Scece Vol. 3 9 o O a Trucated Erlag Queug Sytem wth Bul Arrval Balg ad Reegg M. S. El-aoumy ad M. M. Imal Departmet of Stattc Faculty Of ommerce Al- Azhar Uverty. Grl Brach Egypt mmmal_eg@yahoo.com Abtract The am of th paper to derve the aalytcal oluto of the queue:m x /E /I/ wth balg ad reegg whch ) Ut arrve batche of radom ze wth the terarrval tme of batche followg egatve expoetal dtrbuto. ) The queue dcple FFS t beg aumed that the batche are preordered for ervce purpoe ) The ervce tme dtrbuto Erlaga wth K tage.recurrece relato coectg the varou probablte troduced are foud. Some meaure of effectvee a L ad Lq are deducted ad ome pecal cae are alo obtaed.. Itroducto May reearcher a ox ad Mller [3] robhu [9] ad Saaty [] tuded Marova queue wth bul arrval but they dd ot tae coderato of balg ad reegg. Some reearcher a Bur [] tuded the queue: M x /M/ the homogeeou cae wth bulg ad delay. Other rome et al. [4] dcued the queue: M x /M/ Abou-ELAta et al. [] dcued wth the bul arrval queue M x /M/ wth both cocept of balg ad reegg ad the o-trucated queue: M x /E / had bee olved by Medh [7] who obtaed the oluto term of geeratg fucto. Th paper amed to treat the bul arrval queue for o-marova (Erlaga ervce) queue: M x /E // wth balg ad reegg. I th wor we obta (the probablte that there are ut the ytem ad the ut the ervce occupe th S tage (S )) term of by ug a recurrece

2 4 M. S. El-aoumy ad M. M. Imal relato. Alo we obta the probablty of a empty ytem ad deduce ome pecal cae.. Decrpto of the ytem I the preet ytem t aumed that the ut arrval the ytem batche of radom ze X.e. at each momet of arrval there a probablty ( Χ ) that ut arrval multaeouly o ad the terarrval tme of batche follow a egatve expoetal dtrbuto wth tme depedet parameter λ. Let λ Δ t (... ) be the frt order probablty that batch of ut come tme Δ t. The ervce chael buy f a ut foud ay oe of the K tage (of M/E K //). I th cae the arrval ut form a queue wth capacty. oder the bal cocept wth probablty prob.{a ut o the queue} where < f () ad f a umber of ut the ytem. It alo aumed that the ut reege accordg to a expoetal t dtrbuto f ( t) α e α t α > wth parameterα. The probablty of reegg a hort perod of tme Δt a gve by: ( ) Δt for < ad r for r α. Ug the rate out rate approach the teady tate dfferece equato ca be wrtte a follow: For we have λ () μ For we have μ ( μ λ ) ( μ α ) ( μ λ ) λ c () For ()- we have ( μ ( ) α) ( μ ( ) α λ) λ ( μ ( ) α) ( μ ( ) α λ) λ λ (3)

3 O a trucated Erlag queug ytem 5 For we have o λ λ λ α μ λ λ λ λ α μ α μ (4) Summg () over ad ug (4) we have λ λ α μ (5) alo ummg (3) over ug (5) ad addg the reult obtag for ()- we get: θ (6) where: ) (. α μ λ θ From the t equato of (): we have ) (. θ θ (7) Thu upo ug t equato of (3) ad (6) we get for () -. () θ θ θ (8) Alo from t equato of (4) ad equato (6) at θ θ θ θ. (9) Equato (7) (8) ad (9) are requred recurrece relato whch gve all probablte term of whch telf may ow be determed by ug the ormalzg codto o hece all the probablte are completely term of the queue parameter. The followg example llutrate the dcued method.

4 6 M. S. El-aoumy ad M. M. Imal 3. Example I the above ytem: M x /E /I/ ( α ) wth balg ad reegg lettg 3.e. the queue: M x /E /I/4 ( α ) wth balg ad reegg the reult are: a a b b 3 g 3 g 4 f 4 g where: λ θ () μ α ( ) λ μ θ a θ a θ( θ ) { ( a a ) ( )} b θ 3 4 ( θ ) b θ b a { ( b b ) ( )( a a ) } g θ ( θ ) g ( b ) g 3 θ 3 a { ( g g ) ( )( b b ) ( )( a a ) } f θ ( θ ) f { ( g f ) ( b g f ) ( a b g f )( )} f 4 θ form the ormalzg codto: { a a b b g g f f }. Therefore the expected umber the ytem ad the queue repectvely. L L {( a a ) ( b b ) 3( g g ) 4( f f )} L ( ) q ( b b ) ( g g ) 3( f f ) { }. L q 4. Specal cae: - Let Equato ( 7 ) ( 8) ad ( 9 ) the we get the bul queue:

5 O a trucated Erlag queug ytem 7 M Χ /M// ( α ρ ) ad the reult are: ρ ρ c δ ρ ρ δ c 3() η δ ρ c c c ρ δ λ α whch agree wth the reult of El-aoumy M.S. [5] where ρ δ μ μ η ρ c p ρ p ρp q c 3() - If we put c δ we get the ytem: M/E // wth balg ad reegg 3- Let K δ c we get the ytem: M/M// ( α ) ρ ρ ρ δ ρ 3(). δ ( ) ρ( γ ) Or () ( δ ' ) γ ρ ρ ( δ ' ) λ μ where γ δ ' ( δ ') δ '( δ ' )...( δ ' ) ad ( δ '). α α 4- Moreover let ad α we have the ytem; M/M// wthout bulg or reegg whch were tuded by Whte et al. [] Gro ad Herr [6] More [8] ad other. The effect of λ ad α o L ad L q are howg by table () to (5) for μ

6 8 M. S. El-aoumy ad M. M. Imal Table ():- α. λ L L q L L q L L q L L q L L q L L q L L q L L q L L q L L q

7 O a trucated Erlag queug ytem 9 Table ():- α.3 λ L L q L L q L L q L L q L L q L L q L L q L L q L L q L L q

8 M. S. El-aoumy ad M. M. Imal Table (3):- α.5 λ L L q L L q L L q L L q L L q L L q L L q L L q L L q L L q

9 O a trucated Erlag queug ytem Table (4):- α.8 λ Β. L L q L L q L L q L L q L L q L L q L L q L L q L L q L L q

10 M. S. El-aoumy ad M. M. Imal Table (5):- α. λ L L q L L q L L q L L q L L q L L q L L q L L q L L q L L q ocluo I th paper we derve a exact all recurrece relato to tudy the teadytate probablty that there are ut the ytem ad the ut the ervce beg the th tage. We alo derve aalytcal expreo for teady- tate ytem performace meaure a L ad Lq term of o whch ca be computed ug the ormalzg codto. Some umercal reult are gve Table () - (5) for pecal cae. From the umercal reult t clear that L ad L q are creag a λ ad are creag.

11 O a trucated Erlag queug ytem 3 Referece [] Abou-El-Ata M.O. ad Al-Seedy R.O. 989 "The bul arrval queue: M Χ /M/ wth bulg ad reegg" Joural of the Faculty of Educato A Sham Uverty o.4. [] Bure.J. (975) Delay gle-erver queue wth batch put Op Re [] 3 ox D. R. ad Mlle H. D. (965) The Theory of Stochatc rocee Secto 4 Wley ew Yor. [] 4 rome M.V. haudhry M.L. ad Grama W.K. 979 "Further reult for the queug ytem M Χ /M/c ". Joer. Re. Soc. o [] 5 El- aoumy M. S. (3) A Study of The Trucated Hyper Expoetal Queue. h. D. The Zagazg Uverty- Baha. [] 6 Gro D. ad Harr. M. (974) Fudametal of Queug Theory. Joh Wley ew Yor. [] 7 Medh J. 984 "Recet Developmet Bul Queug Model ".Wley Eater Lmted ew Delh [] 8 More.M. (958) Queue Ivetore ad Mateace. Joh Wley ew Yor. [] 9 robh. U. (965) Queue ad Ivetore Wley ew Yor. [ ] Saaty T. L. (96) Elemet of Queug Theory Secto 7.5 McGraw-Hll ew Yor. [ ] Whte J.A.; Shmdt J.W. ad Beett G. K. (975) Aaly of Queug Sytem. Academc pre ew Yor. Receved: September 8