Meenu Gupta, Man Singh & Deepak Gupta

Transcription

1 IJS, Vol., o. 3-4, (July-December 0, pp Serals Publcatons ISS: X THE STEADY-STATE SOLUTIOS OF ULTIPLE PARALLEL CHAELS I SERIES AD O-SERIAL ULTIPLE PARALLEL CHAELS BOTH WITH BALKIG & REEGIG DUE TO LOG QUEUE AD SOE URGET ESSAGE eenu Gupta, an Sngh & Deepak Gupta Abstract: Ths paper consders the most approprate & more general ueung model n respect of customers whch are allowed to leave the system at any stage wth or wthout gettng servce. The paper consders the steady-state behavor of the ueung processes when seral multple parallel servce channels havng balkng & renegng, are lnked wth non-seral multple parallel channels wth balkng & renegng phenomenon, wheren: Each of seral servce channels & non-seral servce channels have dentcal multple parallel channels. Posson arrvals & exponental servce tmes are followed. The servce dscplne follows SIRO rule (servce n random order nstead of FIFO rule (frst n frst out. The customer becomes mpatent n the ueue after sometme and may leave the system wthout gettng servce and may renege due to urgent message also. Watng space s nfnte. Keywords: Posson stream, Renegng, Balkng, Traffc ntensty, Steady-state, Parallel channels.. ITRODUCTIO The problem of seral ueues was studed by O Bren (954, Jackson (954, Barrer (955, Hunt(955 and aggu (970 n steady-state wth Posson assumptons wth the restrctons that the customer must go through each servce channel before leavng the system. Sngh (984 studed the problem of seral ueues ntroducng the concept of renegng. The steady-state solutons of multple parallel channels n seres wth mpatent customers are obtaned by Sngh & Ahua (995. The solutons of seral and non-seral st Internatonal Conference on athematcs and athematcal Scences (ICS, 7 July 0.

2 490 eenu Gupta, an Sngh & Deepak Gupta ueung processes wth renegng and balkng phenomenon have been studed by Vkram& Sngh (998. The steady-state soluton of seral and non-seral ueung processes wth renegng and balkng due to long ueue and some urgent message and feedback phenomenon have been obtaned by Sngh, Punam & Ashok (009. In our present socety, the mpatent customers generate the most approprate and modern models n the ueung theory. Incorporatng ths concept, we study the steady-state analyss of general ueung system n the sense that: servce channels n seres are lnked wth non-seral channels both havng renegng and balkng phenomenon where each of seral servce channels & non-seral servce channels have dentcal multple parallel channels. The nput process s Posson and the servce tme dstrbuton s exponental. The servce dscplne follows SIRO-rule(servce n random order nstead of FIFO- rule (frst n frst out The customer becomes mpatent n the ueue after sometme and may leave the system wthout gettng servce and may renege due to urgent message also. The nput process depends upon the ueue sze n seral and non-seral channels. Watng space s nfnte.. FORULATIO OF ODEL The system conssts of Q ( =,,, seral servce phases where each servce phase Q has c ( =,,, dentcal parallel servce facltes and Q non-seral servce phases ( =,,, where each servce phase Q has d ( =,,, dentcal parallel servce facltes wth respectve servers S ( =,,, and S ( =,,,. Customers demandng dfferent types of servce arrve from outsde the system n Posson dstrbuton wth parameters ( =,,, at Q servce phase and ( =,,, at Q servce phase respectvely. But the sght of long ueue at Q and Q, may dscourage the fresh customers from onng the system and may decde not to enter the servce channels Q ( =,,, and Q ( =,,, then the Posson nput rates ( =,,, and ( =,,, would be n and m where n and m are the ueue szes of Q and Q. Further, the mpatent customers onng any seral or non-seral servce channels Q or Q may leave the ueue wthout gettng servce after a wat of certan tme. The customer may also leave any servce channel wthout gettng servce f they receve urgent call whle watng. The servce tme dstrbuton for the servers S ( =,,, and S ( =,,, are mutually ndependent negatve exponental dstrbuton wth parameters ( =,,, and ( =,,, respectvely. After the completon of

3 The Steady-State Solutons of ultple Parallel Channels n Seres servce at Q ( =,,,, the customers ether leave the system wth probablty p or on the next phase wth probablty n (,,..., n such that p n ; ( =,,,. After completon of servce at Q, the customers ether leave the system wth probablty p or ons any of the Q ( =,,, wth probablty m ( =,,, such that p m. If the customers are more than c n the Q servce phase, all the c servers wll reman busy and each s puttng out servce at the mean rate and thus the mean servce rate at Q s c, on the other hand f the number of customers s less than c n the Q servce phase,only n out of the c servers wll be busy and thus the mean servce rate at Q s n ( =,,,. Further, f the customers are more than d n the Q servce phase, all trhe d servers wll reman busy and each s puttng out servce at mean rate µ and thus the mean servce rate at Q s d µ, on the other hand f the number of customers s less than d n the Q servce phase, only m out of d servces wll reman busy and thus the mean servce rate at Q s m µ ( =,,,. It s assumed that the servce commences nstantaneously when the customer arrves at an empty servce channel.. Formulaton of Euatons Defne P (n, n,, n ; m, m, m 3,, m ; t as the probablty that at tme t, there are n customers (whch may renege or after beng servced by the Q phase ether leave the system or on the next servce phase watng n the Q servce phase ( =,,,, m customers (whch may renege or after beng servced leave the system watng before the servers S ( =,,,. We defne the operators T. and T. and T., +. to act upon the vector n ~ = (n, n,, n and T. and T. and T., +. to act upon the vector m ~ = (m, m,., m as follows: T. (n ~ = (n, n,, n,, n T. (n ~ = (n, n,, n +,, n T., +. (n ~ = (n, n,, n +, n +,, n T. (m ~ = (m, m,, m,, m T. (m ~ = (m, m,, m +, m T., +. (m ~ = (m, m,, m +, m +.,, m Followng the procedure gven by Kelly (979, we wrte dfference-dfferental euatons for n 0, m 0, ( =,, 3,, ; ( =,, 3,,.

4 49 eenu Gupta, an Sngh & Deepak Gupta n ( { ( } m n n n c r n dp ( n, m; t P( n, m ; t dt ( m { m ( } m d R m P ( T ( n, m ; t P( n ; T ( m ; t ( n. m. ( r P( T ( n, m ; t n c n. ( n ( n.. ( n. P ( T ( n, m ; t p P( T ( n, m ; t P( n, n,..., n, T ( m ; t ( n ( m. ( ( R P( n ; T ( m ; t ( ( m m d ( m. Where (x = when x 0 0 when x 0 (n c = 0 when n c when n c m d = 0 when m d when m d n = m = n = m when m d d when m d when n c when n c n n = m = n when n c c when n c n m when when when n c n c when m d m d r n = R m = T0 n e T ; =,,..., 0 n ( e T0 m e T ; =,,..., 0 m ( e

5 The Steady-State Solutons of ultple Parallel Channels n Seres Where r n and R m are the rates at whch the customers renege after a wat of certan tme T 0 and T 0 whenever there are n and m customers n the Q and Q servce phases respectvely. n, m, n, m are the arrval rates and servce rates respectvely and are the mean renegng rates at seral and non seral multple parallel servce phases due to some urgent message and P (m ~, n ~ ; t = 0 f any of the arguments s negatve.. Steady-State Euatons We wrte the steady-state euatons of the ueung model by euatng the tme dervatve to zero n the euaton ( n ( { ( } m n n n c r n P ( n, m ( m { m ( } m d R m P ( T ( n, m ; t P( n ; T ( m ( n. m. ( r P ( T ( n, m n c n. ( n ( n.. p P( T ( n, m ( n. P( T ( n, m P ( n, n,..., n, T ( m ( n ( m. ( ( R P ( n ; T ( m ( ( m m d ( m. For n 0, m 0; ( =,,, ; ( =,,,. Case (: When n < c and m < d p + = ( =,, 3,..., and p = ( =,, 3,...,

6 494 eenu Gupta, an Sngh & Deepak Gupta The resultng euatons ( n ths case reduce to as under: ( n { n } ( m { m } P ( n, m P ( T ( n, m P ( n ; T ( m ( n P ( T ( n, m.... p ( n P( T ( n, m ( n P( n, n,..., n, T.( m.. ( m P( n ; T ( m (3 The solutons of the steady-state euatons can be verfed to be: n n n3 3 P ( n, m P (0, 0 n n n 3 3 n m... n m m m... m m For n 0, m 0; ( =,,, ; ( =,,,. Where = (4 = k = k + k k ; k =, 3,...,. Case (: When n c and m d wth the condtons p = ; ( =,,..., n and p = ; ( =,,..., m

7 The Steady-State Solutons of ultple Parallel Channels n Seres the resultng euatons ( wll reduce to as under: ( n ( c rn n m P( n, m ( m { d ( Rm } P ( T ( n, m P ( n ; T ( m ( p c r P ( T ( n, m.. ( n. n m c P T (... n m ( m. ( n, m c P( n, n,..., n, T ( m ( d R P ( n ; T ( m (5 The solutons of the steady-state euatons (5 can be verfed to be: { ( c r c } P( n, m P(0,0 n n ( ( n n n n n ( c r n ( c r ( c r( n n3 { 3 ( c r ( n } c n3 n3( c33 r3 3 ( c r ( n n { ( c r ( n } c n n( c r ( c r ( n I n3... m m ( c ( c m m m ( d R m ( d R m ( c... m m ( d R (6 n

8 496 eenu Gupta, an Sngh & Deepak Gupta Where ( c r ( n c m ( n ( c r ( c r ( n ( n n k k ( cu r ( n k k uk ck k ; k, 3,..., ( n k We obtan P (0 ~, 0 ~ from (4 and (6 by the normalzng condton m 0 n 0 P ( n, m and wth the restrctons that traffc ntensty of each servce channel of the system s less than unty Thus P (n ~, m ~ s completely determned. REFERECES [] Hunt G. C., (955, Seuental Arrays of Watng Lnes, Opps. Res., 4: [] Jackson R. R. P., (954, Queung Systems wth Phase Type Servce, Operatons Research Quarterly, 5(: [3] Kelly F. P., (979, Reversblty and Stochastc etworks, John Wley and Sons, Inc., ew York. [4] aggu P. L., (970, On Certan Types of Queues n Seres, Statstca eerlandca, 4: [5] O Bren G., (954, The Soluton of Some Queung Problems, J. Soc. Ind. Appl. ath., : 3 4. [6] Sngh an, (984, Steady-State Behavor of Seral Queung Processes wth Impatent Customers, ath. Operatons forsch, U. Statst. Ser. Statst., 5(: [7] Sngh an, and Ahua Asha, (996, The Steady State Solutons of ultple Parallel Channels n Seres wth Impatent Customers, Intnl. J. gmt. Sys., (: [8] Sngh an Punam, and Ashok Kumar, (008, Steady-State Solutons of Seral and on-seral Queung Processes wth Renegng and Balkng due to Long Queue and Some Urgent essage and Feedback, Internatonal Journal of Essental Scence, (: 7.

9 The Steady-State Solutons of ultple Parallel Channels n Seres [9] Vkram, and Sngh an, (998, Steady State Soluton of Seral and on-seral Queung Processes wth Renegng and Balkng Phenomenon, Recent Advances n Informaton Theory, Statstcs and Computer Applcatons,CCS Haryana Agrcultural Unversty Publcaton, Hsar eenu Gupta Asstt. Prof., Dept. of athematcs, Dr. Bhm Rao Ambedkar Govt. College, Kathal-36 07, Haryana, Inda. E-mal: meenugupta0@gmal.com an Sngh Prof. (Retd., CCS, Haryana Agrcultural Unversty, Hsar, Haryana, Inda. Deepak Gupta Prof. & Head, Dept. of athematcs,.. Engg. College, ullana, Ambala, Inda.