On The Queuing System M/E r /1/N

Transcription

1 On The Queuing System M/E //N Namh A. Abid* Azmi. K. Al-Madi** Received 3, Mach, 2 Acceted 8, Octobe, 2 Abstact: In this ae the queuing system (M/E //N) has been consideed in equilibium. The method of stages intoduced by Elang has been used. The system of equations which govens the equilibium obabilities of vaious stages has been given. Fo geneal N the obability of stages of sevice ae left in the system, has been intoduced. And the obability fo the emty system has been calculated in the exlicit fom. Key wods: Queuing system, Elangian sevice, steady-state. Intoduction: The queuing oblems in which the sevice-facility consists of a numbe of sevice channels in seies have been studied. But only a few have laced the estiction of a finite queue size. One of the leading investigations in this diection was made by Hunt []. He obtained the maximum ossible utilization of the system the exected numbe of customes in the system, assuming exonential sevices. His esults have athe been limited in chaacte. Hollie Boling [2] esented some new extensions of Hunt s wo. Moe secifically they studied the queuing system which consists of N sevice channels in seies, each channel has eithe an exonential o Elangian sevice-time. They obtained the steady state mean outut ate also mean numbe of customes in the system. Recently many authos have consideed the system M/E /. Giffiths his colleagues [3], have obtained the tansient hase obabilities in tems of a new genealization of the modified Bessel function, mean waiting time in the queue. aoumy [4], has deived the analytic solution of the tuncated Elangian sevice queue with statedeendent ate, baling eneging. Shawy [5] has obtained the analytical solution of the queue M/E / //N fo machine intefeence system with baling eneging. oudavish [6], has obtained, the steady state n s obabilities with the unit in the sevice being at stage s of the tuncated Elangian sevice queuing system with state-deendent with fuzzy aival ate. It is nown that when we use distibution othe than exonential, the memoyless oety of the exonential will be lost the analysis becomes moe comlicated. Secial methods in this case have been devised. Some of these secial ones ae method of imbedded-maov chain, [7], the method of Lindley's integal equation [8]. These methods do not in geneal give an exlicit solution but give some functions associated with the solution. In this ae we conside the Elangian queuing system (M/E //N) in equilibium, whee customes aive at om at mean ate λ the *Baghdad Univesity, College of Science fo Women, Deatment of Mathematics namh_abed@yahoo.com *Al-Zaytoonah Univesity,College of Science, Deatment of Mathematics d_azmi_almadi@yahoo.com

2 sevice times have an Elangian distibution with aamete mean sevice ate μ. Even though thee is a single -stage Elangian seve, we can conside this sevice to mae u of exonential sevices in seies, each with a mean sevice ate μ. That means Elang has etained the valuable oety of exonential still allowing fo moe geneal distibution. This consideation will hel in fomulating the equilibium. obabilities equations with the hel of state tansition ate diagam by using the insection method. To give a ecise descition of the state of the system, let then () customes in the system out of which (-) ae in the queue the one in sevice is in the stage of sevice yet to be comleted. If () denotes the numbe of stages of sevice left in the system at this moment, then = (-)+(-i+) = -i+, i=, 2,...,, =, 2,..., N+ () Thee is secial state = which is not coveed by () which means that the system is comletely emty. the total numbe of ossible states, is theefoe (N+) + which can be designated by the symbols E, E, E 2,.., E (N+). We shall use the following notation fo equilibium obabilities. = obability stages of sevice ae left in the system, (2) = obability thee ae customes in the system (3) with these definitions it is now clea that, the obability fo the emty system, is same as the obability of zeo stages of seves ae left in the system,=,2,,n+ (4) ( ) Equations fo Equilibium obabilities: Let us stat in this section by dawing the state tansition ate diagam fo (M/E //N), to hel us fo witing the equilibium equation fo state obabilities: Fig. () State tansition ate diagam Since we ae consideing only the equilibium of the system, the ates at which obabilities flow into a state must balance with the obabilities which flow out fom that state. Thus, it is clea that fom fig. we have the following equations, (5) ( ) i i,i=,2,,- (6) ( ) i i i,i=, +,,N (7)

3 , i=n+, i i i N+2,,(N+)- (8) ( N ) N, (9) The equations (5) to (9) ae in fact (N+)+ equations in as many unnown which ae state obabilities. These equations ae all linea homogenous, they ae consistent. These equations ae such combination of the othes. So in essence they ae only (N+) unnowns in tems of one of them, say, this last unnown should then be detemined with the hel of the consevation equation, viz., ( N) i () i Obtaining will immediately give all the othe obabilities. Once this has been done we can then find any othe desied statistical oety of the queuing system. Let us define q, () ewite the equations (5) to (9) as below: ( q ), (2) q, i=,2,.,- (3) i i i qi ( q ) i,i=,+,...,n (4) i i ( q ) i, i=n+, N+2,,(N+)- (5) ( N) ( q ) N (6) Now, we shall study the secial case N=, i.e., no queue-length is allowed. In this case the state tansition-ate diagam educes to Fig. (2) State tansition ate diagam fo N= the equilibium equations fo state obabilities ae i i, i=,2,..,- (7) And the last state E gives the equation ( q ) (8) Solving these equations one can have 2... (9) ( ) Using the consevation equation () one can easily show that the obability of emty system is (2) Now we ae in the osition to give ou main statements thei ove. In the fist theoem, we give the obability of stages of sevice ae left in the system. Theoem () Fo fixed N, whee N is the numbe of customes ae emitted in the queue of the system, the obabilities of stages of sevices ae left in the system is m ( m ) i ( m ) i h m i c ( ) ( ) q h h h h ( m ) i h whee c ( q ), m=,,2,,n-, i=,2,..., oof Using equation (2) in equation (3) ecusively, one can find that:

4 i i cq, i=, 2,,.(2) Maing use of (2) ecusively using the fist (+) equations fom (4) we can get c q iq i q i i i 2 i ( ), i=,2,,+ (22) Now using the esults in (2) (22) ecusively using the next + equations fom (4) one can have ( q ) ( q )... ( q )N N 2 ( q ) 2 q 2 Thus q 2 N 2 ( q ) Note that at this oint we aly the following chec fo validity of ou esult. utting = theefoe q = ρ +. Hence i i i i 2 2 i c q i q i q q i q q i i i 2 i 2 i 2 i 3 i=2,3,.,+2 (23) If we continue in this way using induction on (i) fo fixed N, esult follows. Theoem (2) The obabilities fo stages N+2,... (N+)- ae left in the system ae... N2 2 N And this is the ight value of in the case of (M/M//N) queue to which the esent queue (M/E //N) degeneates. N h ( N ) h ( N ) ih N i c ( ) ( ) ( N ). q ( N ) i. q h h h ( ) N i q in ( ) N q N Whee c ( q ) i=2,,-. oof: By using the same ocedue as in theoem (), the esult follows. Calculation of We oceed to calculate, using the consevation equation, that means 2... N N Since 2 ( ) ( ) 2... ( ) N N q N q N q. Theefoe N N ( ) q Conclusion In this ae the queue (M/E //N) has been consideed we have found the obabilities of stages of sevice ae left in the system we calculate the obabilities fo the emty system, with this value of we can calculate all the desied equilibium state obabilities, any statistical oeties of the system. Refeences:. Hunt, G.C "Sequential Aays of Waiting Lines" Oeations Reseach. 4: Hollie, F. S. Boling, R. W "Finite Queues with Exonential o Elang sevice Times, A Numeical Aoach", Oeation Reseach. 5:

5 3. Giffiths, J. D. Leoneno G. M. Williams J. E. 26. "The Tansient Solution of M / E /", Oeations Reseach Lettes. 34(3): El-aoumy, M.S. 28. "On a Tuncated Elangian Queueing System with State-Deendent Sevice Rate, Baling Reneging" Al. Math. Sci. 2(24): Shawy, A.I. 25. "The Sevice Elangian Machine Intefeence Model: M/E///N with Baling Reneging", J. Al. Math. & Comuting. 8(): oudavish, A. 2. "Tuncated Elangian Queueing system with State-Deendent Sevice, Fuzzy Aival Rate with Baling Reneging" Fouth Intenational Confeence on Neual, aallel Scientific Comutations, Moehouse College, Atlanta, Geogia, USA. 7. Kendall, D. G "Stochastic ocesses Occuing in the Theoy of Queues Thei Analysis by the Method of Imbedded Maov- Chain", Ann. of Math. Stat. 24: Lindly, D.V "The Theoy of Queues with a Single seve", oc. Camb. hill. Soc. 48: حول نظام الطوابير M/E N// نعمة عبدهللا عبد* عزمي قاسم الماضي** *جامعة بغداد كلية العلوم للبنات قسم الرياضيات. ** جامعة الزيتونة االردنية كلية العلوم قسم الرياضيات. الخالصة: في هذا البحث تمت دراسة نظام الطوابير )N// )M/E في حالة االستقرار لعدد محدد )N( من الزبائن في الطابور. وتم احتساب احتمالية من مراحل الخدمة المتبقية في النظام ومن ثم احتساب احتمالية ان يكون النظام خاليا من الزبائن وقد اعطيت النتائج بصيغة صريحة. 733