The Non-Truncated Bulk Arrival Queue M x /M/1 with Reneging, Balking, State-Dependent and an Additional Server for Longer Queues
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1 Alied Maheaical Sciece Vol. 8 o The No-Tucaed Bul Aival Queue M x /M/ wih Reei Bali Sae-Deede ad a Addiioal Seve fo Loe Queue A. A. EL Shebiy aculy of Sciece Meofia Uiveiy Ey elhebiy@yahoo.co Abac The ai of hi ae i o deive he oluio of he o-ucaed queue: M x /M/ wih eei bali ae-deede ad a addiioal eve fo loe queue. I hi cae he ui aive i bache of ie X which i a ado vaiable. A he ed of hi ae oe ecial cae ae deduced. Keywod: No-ucaed queue loe queue Bali Reei coce Pobabiliie eeai fucio Ioducio May eeache udied he oble of bul aival queue bu wihou ay coce. Soe eeache udied he queue: M x /M/ i he hooeeou cae wih bloci ad delay. Ohe dicued he queue: M x /M/C ad M x /G/. Abou-El-Aa e al [] dicued he bul aival queue: M x /M/ wih boh coce of bali ad eei. El-Paouy [4] dicued he ucaed queue M x /M//N wih he ae coce. I hi ae i i aied o ea he oucaed bul aival queue: M x /M/ wih bali eei ae-deede ad a addiioal eve fo loe queue. The dicilie coideed i he uual oe fi i fi ou IO. I hi wo he eeache iveiae he obabiliy eeai fucio of he ube of ui i he ye. The oe ecial cae ae deduced.
2 748 A. A. EL Shebiy The equilibiu diibuio Coide he ieaival ae of he ui be a exoeial diibuio wih ae. The evice ie ae i alo exoeially diibued wih ae. The ui ae eved accodi o he dicilie IO. Aue he ou ie i a ado vaiable wih diibuio: X ρ wih ea: ad vaiace: σ Ad he bal coce wih obabiliy - i.e.: ob.{a ui joi he queue} > ad fo Alo coide he eei coce which ea ha a ui ay eee wih fucio afe joii he queue fo evice fo a ceai ie which i a ado vaiable wih a obabiliy deiy fucio: f e > Le: - if ui ae i he ye ad The evice ie ae i cae of ae-deede ad a addiioal eve fo loe queue i a follow: 3 Now le be he obabiliy ha hee ae ui i he ye a ie by: Theefoe he diffeeial-diffeece equaio ae: { N } N [ ]
3 No-ucaed bul aival queue 749 ] [ A he eady-ae diffeece equaio ae 3 4 ] [ 5 6 Le u alo defie he followi wo obabiliie eeai fucio:
4 75 A. A. EL Shebiy Ρ G ad 7 Mulilyi elaio 4 ad 6 by ui ove ad addi elaio 3 ad 5 we e: Ρ Ρ 8 ; whee: K ad G 9 Relaio 8 i a fi ode diffeeial equaio i P. A i i clea ha: G P. ; E ad G Ρ E The oluio of elaio 8 i: Ρ c d. Whee: Δ Δ d ad e ; Bu a P C d Ρ 3 Soe ecial cae - Cae: le he we obai hi queue wih a addiioal eve fo loe queue oly uch ha: ad whee ad
5 No-ucaed bul aival queue 75 - Cae: Le we e hi queue wih bali ad eei oly uch ha: ad Which ae he ae eul a i Abou-El-Aa e al []. 3- Cae: Le we obai he ile-chael queue: M x /M/ wih bali oly. Ad hu fo elaio 8 ad we e: Ρ [ ]. 4 Thee eul ae a i [4] whe N 4- Cae: Le we obai he ile-chael bul queue: M x /M/ oly. Ad hu elaio 4 becoe: Ρ ρ ρ 5 Which ae he ae eul a i Hai [] 5- Cae: Le ad he ui ae aived accodi o he eoeic diibuio: Thu we e ohewie G Δ l
6 75 A. A. EL Shebiy A i Dwih [3] elaio 8 becoe Ρ!!!! d 6 Exce fo he cae: v he l. 7 Refeece [] M.O. Abou-El-Aa ad R.O. Al-Seedy. "The bul aival queue M/M/ wih eei ad bali". Joual of he faculy of educaio Ai Sha uiveiy No [] G. Doald ad G.M. Doald. "udaeal of queui heoy".new Yo Joh Wiley ad So 974. [3] H.B. Doald. Table of ieal ad ohe Mah. daa" Mac. Co. N.Y 96 [4] M.S.M. El-Paouy. "A udy o he ucaed queue".thei ubied o faculy of ciece Meofia uiveiy 995. Received: July 7
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