A Performance analysis of cost and profit model queueing system using Probability Generating function
A Markovian line with changed get-away and state subordinate entry rate has been examined. The likelihood producing capacity of the line estimate at a subjective time is acquired utilizing advantageous variable system. Some significant presentation measures like expected line estimate, anticipated length of inert and occupied period is additionally acquired. We utilized likewise Probability creating capacity technique to infer the unfaltering state probabilities, utilizing which different framework execution estimates that can be gotten.
A cost and profit model is created to decide the ideal number of servers. Under the ideal working conditions, numerical outcomes are introduced in which a few framework execution measures are assessed.
The following notations are used to develop the queue size distribution. Let X be the group size random variable of the arrival, X be the Poisson arrival rate when the server is busy and ?0 be the Poisson arrival rate when the server is on repair, km be the probability that m customers arrive in a batch and G(z) be its probability generating function.
Let S1x and R1xbe the cumulative distributions of the service time and repair time, respectively. Let S(x) and R(x) be the probability density functions of service time and repair time, respectively. S1*t denotes the remaining service time of a batch at an arbitrary time t and R1*t denotes the remaining vacation time of a server at an arbitrary time t.
Let St and Rt denote the Laplace- Stieltjes transforms of S and R, respectively. Tst and Twt are the total customers under service and total customers in the waiting, respectively.
M(t) = 0, if the server is bulk service
= 1, if the server is on repair.
= 2, if the server is on epoch
-ddxpi0x=-?pi0x+m=abpi00sx+i=1MQ1i0sx+?n=0a-1Tigi-nsx, a?i?b (1)-ddxpijx=-?pijx+ ?n=0jpij-kxkm a?i?b-1, j?1 (2)
-ddxpbjx=-?pbjx+ m=abpm b+j0sx+i=imq1b+j0sx+n=0a-1Tn?gb+j-nsx (3)
0= – ?0T0+qm00 (4)0= – ?0T0+qmn0+k=1bTn-k ?0gk, 1?n?a-1 (5)-ddxq10x=-?0q10x+m=abpm00vx (6)-ddxq1nx=-?0q1nx+m=abpmn0vx+k=1nq1n-kx?0km, 1?n?a-1 (7)-ddxq1nx=-?0q1nx+k=1nq1n-kx?0km, n?a (8)-ddxqjnx=-?0qjnx+qjn-10r(x)+k=1nqjn-kx?0km, n?a (9)Where 1?n?a-1, 2?j??m-ddxqjnx=-?0qjnx+k=1nqjn-kx?0gk, n?a, 2?j??m (10)Taking laplace transform equation from (1) to equation (10), then we get
? pi0?-pi00=?pi0?-m=abpmi0+i=1Mq1i0+?n=0a-1Tngi-n s? (11)? pij?-pij0=?pij?-?k=1jpij-k?km (12)? q10?-q100=?0q10?-m=abpm00r? (13)? q1n?-q1n0=?0q1n?-m=abpmn0r?-?0k=1nq1n-k?km (14)? q1n?-q1n0=?0q1n?-?0k=1jq1n-k?km ;n?a (15) ? qj0?-qj00=?0qj0?-qj-100r ?, 2?j?k (16) ? qjn?-qjn0=?0qjn?-qj-1n?-qj-1n0?0k=1nq1n-k? km (17)? qjn?-qjn0= ?0qjn?-?0k=1nqjn-k0 km, 2?j?K, n?a (18)IV. Queue time distribution:
To obtain the PGF of queue time distribution at an arbitrary time epoch, the we defined as
piz,?=n=0?pin?zn; piz,0=n=0?pin0zn; a?i?b qjz,?=n=0?qjn?zn; qjz,0=n=0?qjn0zn; 1?j?MCz= n=0a-1Cnzn (21)By using equation (21) we get,
?-?0+?0 Xz q1z,?=q1z,0-v? n=0a-1m=abpmn0zn (22) By using equation (22) we get,
?-?0+?0 Xzqjz,?=qjz,0=v?n=0a-1qj-1n0zn 2???jm (23)?-?+? Xzpiz,?= piz,0-s?m=abpmi0+i=1mq1i0+ ?n=0a-1Cngi-n (24)Where a?i?b-1zb?-?+?Xzpbz,?=zb pbz,0-s?m=abpmz,0-j=0b-1pmj(0)zj+ i=1mq1z,0-j=0b-m-1gizj+?CzXz-m=0a-1Cmzmj=1b-m-1j=0b-m-1gizj (25)q1z,0= R?0-?0 Xzn=0a-1m=abpmn(0)zn (26)qjz,0= R?0-?0 Xzn=0a-1qj-1n(0)zn 2?j?M (27)From the equation 24 and 25 we get
piz,0=S?-?Xz m=abpmi(0)+i=1Mq1i(0)+?n=0a-1Cngi-n a?i?b-1 (28)zb-S?-?Xz pbz,0= S?-?Xz m=ab-1pmz,0-m=abj=0b-1pmj0zj+i=1mq1z,0-j=0b-1q1j0zj+?CzXz-m=0a-1Cmzmj=1b-m-1Kjzj (29)
pbz, 0=S?-?Xz f(z)zb-S?-?Xz (30)fz= m=ab-1pmz,0-m=abj=0b-1pmj(0) zj+i=1Mq1z,0-j=0b-1qij(0)zj+ ?CzXz-m=0a-1Cmzmj=1b-m-1kjzj (31) From the above equation, we get
qjz,?=R?0-?0 Xz-R(?)n=0a-1j=1Mqj-n(0)zn?-?0-?0 Xz 2?j?M (32)piz,?=S?-?Xz-S(?)m=abpmi0+j=1Mqij0+?n=0MCnki-n?-?-?Xz (33)pbz,?=S?-?Xz-S?f(z)zb-S?-?Xz?-?-?Xz (35)Let p(z) be the PGF of the queue time at an arbitrary epoch. Then
pz= m=ab-1pmz,0+pbz,0+m=ab-1pmz,0+Cz (36)pz=S?-?Xz-1i=abm=abpmi0+j=1Mqji0+n=0a-1Cnki-n-?-?Xz+ S?-?Xz-1f(x)zb-S?-?Xz-?-?Xz+R (?0+?0X(z)-1n=0a-1m=abpmn(0)zn-?0-?0Xz+R (?0+?0X(z)-1)n=0a-1j=1nqjnzn-?0-?0Xz (37)Pi=m=abpmi, 0 Qi=i=1Mqji0 Di=Pi+Qi (38)Modify the equ. (37) and (38), we have
S?-?Xz-1-?0-?0Xzm=ab-1zb-zm Dm +R (?0+?0X(z)-1S?-?Xz-1-?0-?0Xz +zb-S?-?Xz-?-?Xz n=0a-1DnZn+ ?Cz-?0-?0XzXz-1 (zb-1)+S?-?Xz-1
pz=-?0-?0Xz?i=ab-1(zb-zim=0a-1Cmki-m-?+?Xz-?0-?0Xzzb-S?-?Xz 39V. Performance Measure
At the point when the quantity of repair become ?=? at that point, C(z) will become zero, henceforth using equation (39), then obtain the form in the given below,
pz=S?-?Xz-1-?0-?0Xzm=ab-1zb-zm Dm+R (?0+?0X(z)-1S?-?Xz-1-?0-?0Xz+zb-S?-?Xz-?-?Xz n=0a-1DnZn-?+?Xz-?0-?0Xzzb-S?-?Xz The above P(z) gives the PGF of line length dissemination of an queueing framework with state subordinate landings and repairs. The outcome precisely agrees with the queue length circulation of of Ebenesar Anna bakyam and Udhaya Chandrika [12].
The mean waiting time of the customers in the queue E W( ) can be easily obtained using Little’s formula
The time period from the repair initiation epoch to the busy period initiation epoch is called the idle time period. Let I be the random number for idle period.?j=0,1,2,·a-1 number of customers visit in the system during the idle period.
Ij = 1, j customers visit to the system in the idle period
Ij= 0, otherwiseUsing the above condition on the size of queue at the time of service finish epoch, we have
?j=?j+k=0a-1?kpIj=k=1 , j=1,2,3·..a-1 (40)In the expected length of busy period is inferred which is helpful to locate the general expense of the framework. Utilizing a contingent desire idea the normal length of occupied period is determined as pursues
EBP=E(S)p(j=0) (41)Total Cost Analysis
The all out normal expense of the queueing framework is inferred with the accompanying suppositions:
Ch : Handling expense per client per unit time
Cm : Maintaining expense per unit time
Cr : Reward because of get-away per unit time.
The length of cycle is the aggregate of the inert period and occupied period. Presently, the normal length of cycle, ETC is obtained.
ETC=EIP+E(BP)ETC=I?0 j=0a-1?j+ERj=1MjI-n=0a-1i=0n?ipn-1+pn+1+EBP (42)Conclusion
We determined some significant framework attributes through the key likelihood producing capacity and also derived the PGF of the line length distribution. The model is critical since progressively broad circumstances in reasonable applications are considered in the model.
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