On the Robustness of the Characteristics Related to (M\M\1) ( \FCFS) Queue System Model

Transcription

1 ISSN (ONINE): 5-758, ISSN (PRINT): Volume-5, Iue-3, June 5 International Journal of Engineering and Management Reearch Page Number: On the Robutne of the haracteritic Related to (M\M\) (\FFS) Queue Sytem Model P. K. Gupta and Jaideep Goel Department of Statitic, D.A.V. (P.G.) ollege, Muzaffarnagar, (U.P.), INDIA Foret Department (U.P.), Rana Pratap Marg, ucknow, (U.P.), INDIA ABSTRAT Variou Queue ytem have been analyzed in repect of their characteritic uing the correponding probability ma function (p.m.f.). Traffic intenity (denoted by) defined a the ratio of the arrival rate to the ervice rate, i an important parameter of thi p.m.f. With the advancement in cience and technology over a period, the parameter involved in ueue characteritic cannot be conidered a contant, here it hould be recognized that the invetigator ha coniderable a priori knowledge about the variation in thee parameter. Highlighting the point that the baic ueue ditribution can be updated in repect of prior variation in it mean, the preent tudy deal with the analyi of the Senitive/ Robut character of (M\M\): ( /FFS) ueue ytem model when traffic intenity i conidered a a random variable. ot Benefit Analyi i alo preent in the correponding ituation. Keyword---Queue ytem, Traffic intenity, Robutne, Prior and Poterior ditribution, ompound ditribution I. INTRODUTION Queuing theory i concerned with the tatitical decription of the behavior of ueue with finding, e.g., the probability ditribution of the number in the ueue from which the mean and variance of the ueue length and the probability ditribution of the waiting time for a cutomer, or the ditribution of a erver buy period can be found. In operation reearch problem involving ueue, invetigator mut meaure the exiting ytem to make an objective aement of it characteritic and mut determine how change may be made to ytem what effect of variou kind of change in ytem characteritic would be, and whether in the light of cot incurred in the ytem change hould be made to it. A model of the ueuing ytem undertudy mut be contructed in thi kind of analyi and the reult of the ueuing theory are reuired to obtained the characteritic of the model, and to ae the effect of change uch a the addition of an extra erver or a reduction in mean ervice time. The characteritic of variou ueue ytem have been analyzed by uing there repective p.m.f.. The traffic intenity () defined a the ratio of the arrival rate to the ervice rate, i an important parameter of thi p.m.f. and variou type of ueue characteritic are defined uing thi parameter [,8]. In reliability theory thi ratio i alo known a availability ratio. The ituation become alarming when one i going for ueue characteritic of the model of the ame nature accomplihing the ame tak in varying condition. Obviouly, for overcoming the ituation, it eem tatitical logical to aume variation in traffic intenity repreented by known uitable prior ditribution. In thi regard, on the repeated analyi of variou ueue ytem, we have a trong bae for collecting prior information howing variation in. For example, in a Barber hop where the arriving individual are the cutomer and the barber are the erver, thi intenity may differ in repect of day in a week. Another example i repreented by letter arriving at a typit dek, where the letter i repreented the cutomer and typit repreented the erver. Following the concept ome of the tudie like [, 4,5, 6 and 7] include a vat literature on ome ueue characteritic in the Bayeian framework in which updating the prior with experimental data ha been main concern. However in the Bayeian framework, it hould be recognized that the prior do have an impact on the baic ditribution and therefore, the preent tudy conider the analyi of the robutne of ueue' characteritic in (M\M\):(\FFS) ytem when traffic intenity i updating in repect of prior and pot prior variation. 839 opyright -5. Vandana Publication. All Right Reerved. II. STATISTIA BAKGROUND (a) For developing the procedure, we conider the p.m.f. of a (M\M\): (/FFS) ueue ytem with poiion input

2 ISSN (ONINE): 5-758, ISSN (PRINT): and poiion output i x Px f (x, ) ( ). ;...() where x,,,... ; Here, () denote the probability that there are x unit in the ueue. Thu the random variable x follow geometric ditribution with parameter (-). The mean and variance of X i given by E(x) = /(-) And V(x)= /(-)² (b) The invetigator prior belief about, the parameter repreenting traffic intenity i repreented by a beta ditribution of firt kind with p.d.f. B(u, v) u v g( ) ( )...() where ; u, v With mean = u / (u + v) and variance = u v / (u + v) ( u + v + ) (c) The variation in get neutralized if we conider the compound ditribution of X [Johnon (969)] in view of the ditribution of X in () and that of in (). Thi compound ditribution will be [x] f (x,u, v) f (x, ).g( )d (v).(u) (u v) [x ] the p.m.f. of X in (3) i a well known Invere polya egenbergar ditribution [ Johnon (969)] with V(x)= u v (u + v - ) / (v-) (v- ) Here, ; ( x,,,...)...(3) E(x) = (u)/(v-) and [x] u [u(u )...(u x )] tand for acending factorial of u. et the compound ditribution in (3) be called a updated baic ditribution. (d) Further let X=( x, x...x n ) be random ample of ize n from the population (), then the poterior ditribution of, in repect of it prior in () will be 84 opyright -5. Vandana Publication. All Right Reerved.

3 ISSN (ONINE): 5-758, ISSN (PRINT): Where n i i uv nv. ( ) ( x)...(4) B(u y, n v) y x, here [ x,i,,...n] repreent the number of ucceful operation before the firt nature. i (e) Now, the compound ditribution of X in () taking expectation with repect to poterior ditribution of in (4) give another ditribution of X a under f (x,u, v) f (x, ). ( x).d (n v).(y u) (y u v n) [x] [x ] The p.m.f. in (5) i well known polyaegenbergar ditribution with E(x) = (u + y) / ( n + v - ) and V(x)= ( u + y ) ( n + v ) ( y + u + v + n ) / ( n + v ) ( n + v ) ; ( x,,,...)...(5) et the ditribution in (5) i termed a predictive baic ditribution [Sinha (986)]. In the compounding proce a ued in (5), the variation in get neutralized on taking expectation over the function f(x, ) with repect to. Variation in i repreented by poterior ditribution in (4), coneuently the updated compound geometric ditribution in (5) i interpreted a predictive baic ueue ytem. Thu, in the proce, one get three baic ueue ditribution are given in (), (3) and (5) repectively. Thee ditribution are ueful for analyzing the ueue' characteritic of a model in the following three pecific ituation. (i) When the baic ueue ytem model a given in () i ued in the analyi. Here i treated a contant. (ii) When the updated baic ueue ytem model, a given in (3) i ued in the analyi. Here i treated a a random variable with it prior a given in (). (iii) When the predictive baic ueue ytem model a given in (5) i ued in the analyi. Here, the parametric variation in are repreented by the poterior ditribution in (4) which alo incorporate experimental data X= ( x, x...x n ) or ample information III. QUEUE HARATERISTIS IN THE ASE OF THREE SPEIFI SYSTEM MODE On uing the three ueue ditribution in (),(3) & (5), the ytem ueue' characteritic in the correponding ituation have been obtained in the following three ub ection. 3. Queue haracteritic in the cae of baic ueue ytem model: () The average idle time I(t) for which the ytem remain idel i I(t) = P( there i no unit in the ueue) =(-)time. (6) 84 opyright -5. Vandana Publication. All Right Reerved.

4 ISSN (ONINE): 5-758, ISSN (PRINT): () The expected number of the unit in the ytem, i given by = (3) The expected ueue length, i given by = x.px (7) x ( ) (x ).P x (8) ( ) x (4) The expected length of non empty ueue ( \ > ) = P(an arrival ha to wait) ( ) (5) The probability of minimum ueue ize being n i given by (9) n Q m P(x n )..() (6) Variance of the ueue length, V i given by V () ( ) (7) The onitency of the ueue length, given by it o-efficient of variation and i given by V V...() (8) The utilization of the erver i given by = - P( x = ) =...(3) (9) ot Benefit Analyi of the model : For formulating the cot benefit analyi of the ytem, let u define the profit function a a function of i.e. P ( ) R. [.. 3.I(t)] (4) Where, P () = Profit function for fixed R = Revenue earned per unit average ueue length = Overhead charge = ot or Expenditure per unit average ueue length = o incurred per unit average waiting line length 3 = o incurred per unit average idle time 84 opyright -5. Vandana Publication. All Right Reerved.

5 ISSN (ONINE): 5-758, ISSN (PRINT): Queue haracteritic in the cae of updated baic ueue ytem model: For analyzing the enitive character of the (M\M\): (/FFS) ueue ytem model in repect of it variou characteritic when traffic intenity i conidered a a random varible with it ditribution in (), we obtained the variou ueue characteriic by uing the p.m.f. of x in (3) a follow () The average idle time I(t) for which the ytem remain idel i I(t) = P ( there i no unit in the ueue) = ( E ( ) ) time. (5) () The expected number of the unit in the ytem, x (3) The expected ueue length, i given by (u) x.f (x,u, v) (6) (v ) i given by (x ).f (x,u, v) = x u v (v ) (u v) (4) The expected length of non empty ueue, ( \ >) = P(an arrival ha to wait) [u (v )] v [ ] (u v) (7) (8) (5) The probability of minimum ueue ize being n i given by Q P[x n ] m n [x] x v(u) (u v) [x]...(9) (6) Variance of the ueue length, Vigivenby V E[x E(x)] (v ).(v ) u v( u v )...() (7) The conitency of the ueue length i given by V{x} = V 843 opyright -5. Vandana Publication. All Right Reerved.

6 ISSN (ONINE): 5-758, ISSN (PRINT): v(u v ) u (v ) () (8) The utilization of the erver i given by v (u v) (9) ot Benefit Analyi of the model : For formulating the cot benefit analyi of the ytem,let u define the profit function a a function of i.e....() P ( ) R. [.. 3.I (t)] (3) Where, P() = Profit function for fixed R,,,, 3 are defined in (4). 3.3 Queue haracteritic in the cae of Predictive baic ueue ytem model: Having obtained the predictive baic ueue ditribution in (5), the variou ueue characteriic by uing the p.m.f. of X in (5) a follow. () The average idle time I (t) for which the ytem remain idel i I (t) = [ E ( ) ] time. (4) () The expected number of the unit in the ytem, i given by x.f (x,u, v) x (y u) (n v ) (5) (3) The expected ueue length, i given by (u nx)(u nx ) (x ).f (x, u, v) (n v )(nx u v n) x = (u y)(u y ) (n v )(y u v n) (6) (4) The expected length of non empty ueue, ( / > ) [(y u) (n v )] (v n) [ ] (y u v n) (7) (5) The probability of minimum ueue ize being n 844 opyright -5. Vandana Publication. All Right Reerved.

7 ISSN (ONINE): 5-758, ISSN (PRINT): i given by Q P[x n ] m n [x] (v n)(y u)...(8) [x] (y u n v) x (6) Variance of the ueue length i given by V E[x E(x)] (y u)(v n)[y u v n ] (v n ).(v n )...(9) (7) The conitency of the ueue length i given by V = V (3) (8) The utilization of the erver i given by (n y) (u v n y)...(3) (9) For formulating the cot benefit analyi of the ytem,let u define the profit function in thi cae a a function of i.e. Where, IV. P ( ) P ( ) R. [.. 3.I (t)] (3) = Profit function for fixed and other contant are defined in (4). DISUSSION AND EXAMPE Now there are three baic ueue ytem in (), (3) and (5) repectively. The ueue characteritic in the correponding ituation have been lited in ection 3.. For analyzing the robut character of the ueue characteritic when i conidered a a random variable, we compare thee a given in ub ection 3. with thoe given in ub-ection 3. and 3.3. For introducing tatitical validity in uch comparion, the parametric value in the three pecific ituation i.e. (,u,v) are o choen o that E(X) for the ditribution in (), (3) and (5) are ame. oneuently, the reference point for comparion are elected o a to atify u u y ( ) (v ) (v n ) On uing the initial geometric ditribution, the repective etimate for ueue' characteritic are given in ub ection 3. and with variation in, have been ummarized in table. For aumed value of the parameter of the prior ditribution, i.e. u,v and y, and alo uing the expreion given in ub-ection 3. and 3.3, the etimate for variou ueue' characteritic have been ummarized in table and table 3 repectively a expected 845 opyright -5. Vandana Publication. All Right Reerved....(33) traffic intenity varie randomly. A comparion of the ueue' characteritic obtained in the three ituation clearly reveal their non-robut character when the variation in the traffic intenity in the claical ueue' ytem are upected. In the preent data etup, the value for the ueue' characteritic tend to be uniformly higher when the prior variation are upected. The trend in all the three table clearly highlighted that modified/updated

8 ISSN (ONINE): 5-758, ISSN (PRINT): ueue' characteritic value are uniformly higher when E() i.e., (it expected mean variation). On the other hand, when > E(), thi updated etimate tend to be higher up to a certain point but thereafter updated ueue' characteritic value are uniformly lower. The point at which the trend are revered i not difficult to etimate. On comparing the variation in co-efficient of variation with repect to and E() in table-, and 3, we oberve that the etimate tend to be more and more conitent a either or it expected mean variation increae. For cot benefit analyi, the three ituation are given in euation (4), (3) and (3), on taking R = R./-, = R.3/-, = R.8/-, = R./- and 3 = R.5/- and for variou value of and it prior variation in profit function are alo hown in table, and 3. In market analyi, by analyzing the above trend, in repect of profit analyi, one can eaily make an economic trade-off in a (M\M\) (\FFS) ueue ytem model. REFERENES [] Ackoof,R.. and Saient, M.W.(968), Fundamental of Operation Reearch Wielly Eatern imited, New York, U.S.A. [] Armero,.(985), Bayeian analyi of (M/M/): ( /FFS) Queue Bayeian Statitic,North Hollnd, Amterdam, pp [3] Johnon, N.. and Kotz,S. (969), Dicrete Ditribution John Wiley and Son, New York, U.S.A. [4] K.K.Sharma and Vinit Kumar(999), Inference on (M/M/): ( /FIFO) Queue Sytem Opearch, Vol. 36, No., pp [5] Muddapur, M.V.(97), Bayeian Etimate of parameter in ome ueuing Model Amer. Int. Math, 4,pp [6] P. K. Gupta, Jaideep Goel and R. Kumar (8), On the Robutne of QUEUE haracteritic in Power Supply Sytem Model Int. J. Eential Science, Vol.- (No.-), pp [7] P. K. Gupta, Jaideep Goel and R. Kumar(9), Poteriror Analyi of The Queue haracteritic in Power Supply Sytem Model Bulle. Of Allahabad Mathematical Society, Vol.-4, part-, pp [8] Taha, H.A. (976), Operation Reearch, An Introduction Macmillan, New York, U.S.A. Table- Etimate of the Queue haracteritic of the ytem when Traffic Intenity i onidered a ontant ρ (/>) Q m ( n 3) V V P ( ) opyright -5. Vandana Publication. All Right Reerved.

9 ISSN (ONINE): 5-758, ISSN (PRINT): Table- Etimate of the Queue haracteritic of the ytem when Traffic Intenity i conidered a a u v E( ) Random Variable (/>) Q m ( n 3) V V P ( ) Table-3 Etimate of the Queue haracteritic of the ytem when Traffic Intenity i conidered a a Random Variable with Poterior prior a it Ditribution u v ρ = E( ) y (/>) Q m ( n 3) V V P ( ) opyright -5. Vandana Publication. All Right Reerved.