Department of Mathematics, Institute of Basic Science. Dr. B.R. Ambedkar, University, Agra-282002, India. (Received January 2006, accepted November 2006) ...
Quality Technology & Quantitative Management Vol. 4, No. 4, pp. 455-470, 2007
QTQM © ICAQM 2007
M/Ek/1 Queueing System with Working Vacation Madhu Jain and Praveen Kumar Agrawal Department of Mathematics, Institute of Basic Science Dr. B.R. Ambedkar, University, Agra-282002, India (Received January 2006, accepted November 2006)
__________________________________________________________________________ Abstract: This paper deals with state dependent M/Ek/1 queueing system with server breakdown and working vacation. As soon as the system becomes empty, the server leaves the system and takes vacation for random duration during which it may perform ancillary duty and is called on working vacation. It is assumed that the server may breakdown when it is busy. The vacation duration and the life time of server are exponentially distributed. Both service times in a working vacation and in a busy period are assumed to be Erlangian distributed. Once server starts the service, he continues until all jobs are served. The Chapman Kolmogorov equations are constructed in order to obtain the steady state probability distribution of the number of jobs in the system. The probability generating function is employed to obtain the average queue length and other system characteristics. Numerical experiment is performed to validate the analytical results. The sensitivity analysis has been done to examine the effect of different parameters. Keywords: Generating function, M/Ek/1, queue size, state dependent rate, unreliable server, working
vacation.
_________________________________________________________________________
1. Introduction acation models had been the subject of interest to queue theorists of deep study in recent years because of their applicability and theoretical structures in real life congestion situations such as manufacturing and production, computer and communication systems, service and distribution systems, etc. The most remarkable works done in recent past by some researchers on vacation models include Fuhrmann and Cooper [4], Lee [11, 12], Doshi [3], Takagi [22, 23, 24, 25, 26] and Lee and Lee [13]. Li and Yang [14] considered a single server retrial queue with server vacations and finite population of the customers. Grey et al. [5] studied a multiple vacation queueing model with server breakdown. They obtained queue length distribution by using probability generating function method. Servi and Finn [18] discussed a classical single server vacation model in which a single server works at a different rate rather than completely stopping during the vacation period. This works was motivated and illustrated by the analysis of a WDM optical access network using multiple wavelengths. Madan and Al-Rub [16] analyzed a single server queue with optional server vacations based on exhaustive service. Some known results of the M/D/1 queue have been derived as a particular case. Gupta and Sikdar [6] studied a single server finite-buffer bulk-service queue and single vacation in which the inter-arrival and service times are exponentially and arbitrarily distributed, respectively. Wu and Takagi [31] presented an M/G/1 queue with multiple vacations and exhaustive service discipline such that the server works with different service times rather than completely stopping service during a vacation. He also derived the distributions for the queue size and the system time for an arbitrary customer in the steady state. Baba [2]
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considered a GI/M/1 queue with vacations such that the server works with different rates rather than being idle during a vacation period. Madan and Al-Rawwash [15] analyzed a single server queue with batch arrivals and general (arbitrary) service time distribution. He obtained steady state results in explicit and closed form in terms of the probability generating functions for the number of customers and the average waiting time. Tian and Zhang [28] considered a queueing system with c servers and a threshold type vacation policy. Using the matrix analytical method, they obtained the stationary distribution of queue length and proved the conditional stochastic decomposition properties. Queueing systems with server breakdown and vacation have also been investigated in different frame-works in recent past. Motivated by unreliable server commonly found in computer systems, communication systems and in manufacturing systems and other day to day realistic queueing problems, many authors have analyzed single and multi server queueing systems with server breakdown. An M/Ek/1 queueing model with arrival rate dependent on server breakdown was investigated by Shogan [19]. Alam and Mani [1], Sztrik and Gál [20, 21], Jayaraman et al. [9], Wartenhorst [30], Hsieh and Andersland [7], Jain [8] studied the queueing models subject to random breakdowns. Further studies on this topic were done by Tang [27] and Ke [10]. Wang [29] studied an M/G/1 queue with second optional service and unreliable server. By using a supplementary variable method, the transient and the steady-state solutions for both queueing and reliability measures of interest have been obtained. A queue with working vacations was first analyzed by Servi and Finn [18], who obtained the queue length distribution of the M/M/1/WV queue. Our work is motivated by such works and extends the queue length distribution for M/Ek/1 queue with working vacations by including phase service for both cases when server rendering service to primary job and on working vacation. The assumption of unreliable server which is subject to breakdown and repair, has also been incorporated. Working vacation is the time for which the server takes vacation from the primary job when there is no job present in the system. During this vacation time, the server may be assigned some other job. The length of time when the server is turned on and in operating condition is known as busy period. The length of time for which the server is broken down and under repair is called the breakdown period. The sum of working vacation, busy period and breakdown period is known as a busy cycle. The analysis is carried out by an analytical method based on generating function. The rest of the paper is structured as follows. The steady state equations have been constructed after stating requisite assumptions and notations in section 2. The probability generating function method is used to determine the queue size distribution and closed form expression of probability of system being empty, in section 3. In the next section 4, we establish various performance characteristics namely the expected queue length. In section 5 the sensitivity analysis is carried out in order to obtain the effect of various system parameters on system performance characteristics. Final section 6 includes the conclusion and further scope of the works.
2. Model Description We consider an M/Ek/1 queueing system with single removable and non-perfect service station which is subject to random breakdowns and repairs. The server works with different service rate rather than completely stopping service during a vacation. For formulation of the mathematical model, the following assumptions are made: Pn,i , j denotes the steady state probability of the system states where n denotes the number of customers in the system (n=0, 1, 2,……), i represents the service phase
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of the customer being served while j describes the state of the service station given as follows:
⎧0, ⎪ j = ⎨1, ⎪2, ⎩
server is on working vacation state server is in busy state server is in breakdown state and under repair
It is assumed that the customers arrive to a single server station according to Poisson fashion with rate dependent upon the server status. The state dependent arrival rate λ of the customers are given as follows:
⎧λ0 , when service station is on working vacation ⎪ λ = ⎨λ1 , when service station is in busy state ⎪λ , when service station is broken down but under repair ⎩ 2 It is assumed that the customers are served in k phases by the server according to Erlang distribution each with mean 1/kμ (in working vacation) and 1/kμ1 (in busy period), respectively. As soon as the system becomes empty, the server goes on vacation in order to attend another job. On returning from vacation if he finds one or more customers waiting in the queue, he starts service of the customers one by one till the system becomes empty, otherwise takes another vacation and so on. The life-time, repair time and the vacation time of the service station follows the exponential distribution with mean 1/α, 1/β and 1/η, respectively. After completion of the repairing, the server provides service with the same efficiency as before breakdown. The long run fraction times for which server is on working vacation, busy and broken down and under repair are denoted by PV, PB and PD, respectively. The expected lengths of working vacation, busy period, breakdown period and the cycle period are denoted by E[V], E[B], E[D] and E[C], respectively.
The system states are denoted by P0,0,0 = Prob. that there is no customer in the system and the server is on vacation Pn,i ,0 = Prob. that there are n customers in the system and the server is rendering ith
(i=1, 2,……,k) phase of service while on working vacation. Pn,i ,1 = Prob. that there are n customers in the system and the server is rendering ith
(i=1, 2,……,k) phase of service when he is in busy state. Pn,i ,2 = Prob. that there are n customers in the system and the customer in service at
the time of breakdown and under repair of server was receiving ith (i=1, 2,……,k) phase service.
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The steady state equations governing the model are
λ0 P0,0,0 = k μ P1,1,0 + k μ P1,1,1
(1)
(λ0 + k μ + η ) P1,i ,0 = k μ P1,i +1,0 ,
i = 1,2,3,........., k − 1
(2)
(λ0 + k μ + η ) P1,k ,0 = k μ P2,1,0 + λ0 P0,0,0
(3)
(λ0 + k μ + η ) Pn,i ,0 = λ0 Pn −1,i ,0 + k μ Pn,i +1,0 , n ≥ 2, i = 1,2,3,........., k − 1
(4)
(λ0 + k μ + η ) Pn,k ,0 = λ0 Pn −1,k ,0 + k μ Pn +1,1,0 , n ≥ 2
(5)
(λ1 + k μ1 + α ) P1,i ,1 = k μ1 P1,i +1,1 + β P1,i ,2 ,
(6)
i = 1,2,3,........., k − 1 k
(λ1 + k μ1 + α ) P1,k ,1 = k μ1 P2,1,1 + β P1,k ,2 + η ∑ P1,i ,0 ,
(7)
(λ1 + k μ1 + α ) Pn,i ,1 = k μ1 Pn,i +1,1 + λ1 Pn −1,i ,1 + β Pn,i ,2 , n ≥ 2 , i = 1,2,3,........, k − 1
(8)
i =1
k
(λ1 + k μ1 + α ) Pn,k ,1 = k μ1 Pn +1,1,1 + λ1 Pn −1,k ,1 + β Pn,k ,2 + η ∑ Pn,i ,0 , n ≥ 2 i =1
(λ2 + β ) P1,i ,2 = α P1,i ,1 ,
i = 1,2,3,........., k − 1
(9) (10)
(λ2 + β ) P1,k ,2 = α P1,k ,1
(11)
(λ2 + β ) Pn,i ,2 = λ2 Pn −1,i ,2 + α Pn,i ,1 , n ≥ 2 , i = 1,2,3,........., k − 1
(12)
(λ2 + β ) Pn,k ,2 = λ2 Pn −1,k ,2 + α Pn,k ,1 ,
(13)
n≥2
3. Probability Generating Function Since there is no possibility to obtain the analytical neat closed expression of P0,0,0 by using recursive technique, the technique of probability generating function can be successful applied for this purpose as detailed below. Define probability generating functions associated with marginal queue size distributions as follows: ∞
X i ( z ) = ∑ z n Pn,i ,0 ,
z ≤ 1, 1 ≤ i ≤ k ,
(14)
Yi ( z ) = ∑ z n Pn,i ,1 ,
z ≤ 1, 1 ≤ i ≤ k ,
(15)
z ≤ 1, 1 ≤ i ≤ k ,
(16)
n =1 ∞
n =1 ∞
Ti ( z ) = ∑ z n Pn,i ,2 , n =1
k
∞
k
GV ( z ) = ∑ X i ( z ) = ∑ ∑ z n Pn,i ,0 , i =1
k
n =1 i =1
∞
k
GB ( z ) = ∑ Yi ( z ) = ∑ ∑ z n Pn,i ,1 , i =1 k
n =1 i =1 ∞
k
G D ( z ) = ∑ Ti ( z ) = ∑ ∑ z n Pn,i ,2 . i =1
n =1 i =1
(17) (18) (19)
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Multiplying (2) and (4) by appropriate powers of z and summing over n, we obtain
X i +1 ( z ) = W ( z ) X i ( z ) ,
(20)
where W ( z ) = [b0 (1 − z ) + c 0 + 1], b0 =
λ0 η , c0 = . kμ kμ
Also W ′( z ) = −b0 ,W ′′( z ) = 0 . Multiplying (1), (3) and (5) by appropriate power of z and summing over n, we get X1 ( z ) = zW ( z ) X k ( z ) + b0 z (1 − z ) P0,0,0 − zP1,1,1 ,
(21)
Similarly multiplying (6) and (8) by appropriate powers of z and summing over n, we obtain
Yi +1 ( z ) = (a1 + 1 − b1 z )Yi ( z ) − rTi ( z ) ,
(22)
where a1 =
(λ1 + α ) λ β , b1 = 1 , r = . k μ1 k μ1 k μ1
Again multiplying (7) & (9), (10) & (12), (11) & (13), respectively by appropriate powers of z and summing over n, we find k
Y1 ( z ) = z (a1 + 1 − b1 z )Yk ( z ) − rzTk ( z ) + zP1,1,1 + η ∑ X i ( z ) , i =1
where b2 = λ2 / k μ1 .
(23)
(b2 + r − b2 z )Ti ( z ) = (a1 − b1 )Yi ( z ) ,
(24)
(b2 + r − b2 z )Tk ( z ) = (a1 − b1 )Yk ( z ) ,
(25)
Equation (20) yields X i ( z ) = W i −1 ( z ) X 1 ( z ) ,
i = 1,2,3,........., k .
(26)
Using equations (21), (26) & (27) and (22) & (24) respectively, we obtain X1 ( z ) =
zb0 (1 − z ) P0,0,0 − zP1,1,1 [1 − zW k ( z )]
⎡ zb (1 − z ) P0,0,0 − zP1,1,1 ⎤ X i ( z ) = W i −1 ( z ) ⎢ 0 ⎥ [1 − zW k ( z )] ⎣ ⎦,
,
i = 1,2,3,........., k ,
Yi +1 ( z ) = V ( z ) Yi ( z ) , where V ( z ) = [1 + a1 − b1 z −
(27)
(28) (29)
r (a1 − b1 ) ] and V(1)=1. b2 + r − b2 z
Equation (29) provide Yi ( z ) = V i −1 ( z ) Y1 ( z ) , i = 1,2,3,........., k .
(30)
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Now substituting the value of Yk ( z ) in (23), we get k
Y1 ( z ) =
zP1,1,1 + η ∑ X i ( z ) i =1 k
1 − zV ( z )
.
(31)
From (30), we have ⎡ zP + η k X ( z ) ⎤ ∑ i ⎢ 1,1,1 ⎥ i −1 i =1 Yi ( z ) = V ( z ) ⎢ ⎥ , i = 1,2,3,........., k . k 1 − zV ( z ) ⎢ ⎥ ⎣ ⎦
(32)
Now from equations (17) and (28), we obtain GV ( z ) =
zb0 (1 − z ) P0,0,0 − zP1,1,1 ⎡ 1 − W k ( z ) ⎤ ⎢ ⎥. 1 − zW k ( z ) ⎣ 1 −W ( z ) ⎦
(33)
It is noticed that k+1 simple zeros (say z = r1, r2, r3,…….., rk+1) outside z = 1 in the denominator are common and cancelled out from both the numerator and the denominator of equation (33), only one factor leaving in the denominator; thus there is the unique root (say r0) inside z = 1 of denominator (cf. Medhi [7], p. 488). Now P1,1,1 = b0 (1 − r0 ) P0,0,0 . Hence equation (33) becomes GV ( z ) =
zb0 (r0 − z ) ⎡ 1 − W k ( z ) ⎤ ⎢ ⎥ P0,0,0 . 1 − zW k ( z ) ⎣ 1 − W ( z ) ⎦
(34)
Now from equations (18) & (32) and (19) & (24), respectively, we obtain GB ( z ) =
zb0 [(1 − r0 )(1 − zW k ( z ))(1 − W ( z )) − η (r0 − z )(1 − W k ( z ))] ⎡ 1 − V k ( z ) ⎤ ⎢ ⎥ P0,0,0 . (1 − zW k ( z ))(1 − zV k ( z ))(1 − W ( z )) ⎣ 1 −V ( z ) ⎦
and GD ( z ) =
(a1 − b1 ) GB ( z ) . (b2 + r − b2 z )
(35)
(36)
Theorem: The probability P0,0,0 of the system being empty is given by
P0,0,0
α ⎡ ⎤ 2 k ⎢ ( β + 1)[{−2 kV ′(1) − k ( k − 1)V ′ (1) − kV ′′(1)}{(η − c 0 )(1 − (c 0 + 1) )} ⎥ ⎢ ⎥ b0 (1 − r0 ) ⎢ −2 kV ′(1){(c 0 + 1) k (c 0 − b0 − η ) + kb0 (c 0 + 1) k −1 (η − c 0 ) + (b0 + η )}] ⎥ = ⎢1 + ⎥ (37) c0 2( −1 − kV ′(1))(1 − (c 0 + 1) k )V ′(1) ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦
Proof: We evaluate the probability P0,0,0 using normalizing condition. For this purpose, we evaluate GV(1), GB(1) and GD(1) from equations (34), (35) and (36), respectively as
M/Ek/1 Queueing System with Working Vacation
GV (1) =
461
b0 (1 − r0 ) P0,0,0 , c0
(38)
b0 (1 − r0 )[{−2 kV ′(1) − k ( k − 1)V ′2 (1) − kV ′′(1)}{(η − c 0 )(1 − (c 0 + 1) k )} GB (1) =
−2 kV ′(1){(c 0 + 1) k (c 0 − b0 − η ) + kb0 (c 0 + 1) k −1 (η − c 0 ) + (b0 + η )}] 2( −1 − kV ′(1))(1 − (c 0 + 1) k )c 0V ′(1) G D (1) =
P0,0,0 ,
α GB (1) , β
(39) (40)
where V ′(1) = −
(λ1 β + λ2α ) , k βμ1
V ′′(1) = −
2λ22α , k β 2 μ1
V ′′′(1) = −
6λ23α . k β 3 μ1
Now using the normalizing condition given by G(1) = GV(1) + GB(1) + GD(1) =1,
(41)
we obtain the value of probability P0,0,0 as given in (37).
4. Performance Characteristics Denote the long run fraction time for which the server is on working vacation, busy and brokendown by PV, PB and PD, respectively. Thus PI = GV (1) =
b0 (1 − r0 ) P0,0,0 , c0
(42)
b0 (1 − r0 )[{−2 kV ′(1) − k ( k − 1)V ′2 (1) − kV ′′(1)}{(η − c 0 )(1 − (c 0 + 1) k )} PB=GB(1)=
−2 kV ′(1){(c 0 + 1) k (c 0 − b0 − η ) + kb0 (c 0 + 1) k −1 (η − c 0 ) + (b0 + η )}]P0,0,0 2( −1 − kV ′(1))(1 − (c 0 + 1) k )c 0V ′(1)
,
(43)
b0 (1 − r0 )α [{−2 kV ′(1) − k ( k − 1)V ′2 (1) − kV ′′(1)}{(η − c 0 )(1 − (c 0 + 1) k )} PD=GD(1)=
−2 kV ′(1){(c 0 + 1) k (c 0 − b0 − η ) + kb0 (c 0 + 1) k −1 (η − c 0 ) + (b0 + η )}]P0,0,0 2 β ( −1 − kV ′(1))(1 − (c 0 + 1) k )c 0V ′(1)
,
Average Queue Length
For brevity of notations, we use a(z)=1−zWk(z), b(z)=1−W(z), c(z)=1−zVk(z), d(z)=1−V(z), e(z)=1−Vk(z), f(z)=1−Wk(z) Denote a = lim z →1 a ( z ) = 1 − (c 0 + 1) k , a ′ = lim z →1 a ′( z ) = −(c 0 + 1) k −1[ −c 0 − 1 + kb0 ] a ′′ = lim z →1 a ′′( z ) = k (c 0 + 1) k −2 b0 [2(c 0 + 1) − ( k − 1)b0 ] a ′′′ = lim z →1 a ′′′( z ) = k ( k − 1)(c 0 + 1) k −3 b02 [ −3(c 0 + 1) + ( k − 2)b0 ] b = lim z →1 b ( z ) = −c 0 , b ′ = lim z →1 b ′( z ) = b0 , b ′′ = lim z →1 b ′′( z ) = 0
(44)
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c = limz →1 c ( z ) = 0, c ′ = limz →1 c ′( z ) = −1 − kV ′(1) c ′′ = limz →1 c ′′( z ) = −2kV ′(1) − k (k − 1)(V ′(1))2 − kV ′′(1) c ′′′ = limz →1 c ′′′( z ) = −3(k − 1)(V ′(1))2 − 3kV ′′(1) − k (k − 1)(k − 2)(V ′(1))3 − 3k (k − 1)V ′(1)V ′′(1) − kV ′′′(1) d = limz →1 d ( z ) = 0, d ′ = limz →1 d ′( z ) = −V ′(1) d ′′ = limz →1 d ′′( z ) = −V ′′(1), d ′′′ = limz →1 d ′′′( z ) = −V ′′′(1) e = limz →1 e ( z ) = 0, e ′ = limz →1 e ′( z ) = −kV ′(1), e ′′ = limz →1 e ′′( z ) =− k (k − 1)(V ′(1))2 − kV ′′(1) e ′′′( z ) = −k (k − 1)(k − 2)(V ′(1))3 − 3k (k − 1)V ′(1)V ′′(1) − kV ′′′(1), f = limz →1 f ( z ) = 1 − (c0 + 1)k , f ′ = limz →1 f ′( z ) = kb0 (c0 + 1)k −1 f ′′ = limz →1 f ′′ ( z ) = − k(k − 1)(c0 + 1)k −2 b02 f ′′′ = limz →1 f ′′′( z ) = k (k − 1)(k − 2)(c0 + 1)k −3 b03 g = e ′′′ + 3e ′′, h = e ′′ + 2e ′, l = 2 f ′ + f ′′, m = f + f ′ Now we derive the expression for the expected number of customers in the system as E[N s ] = E[N v ] + E[N B ] + E[N D ] , where ⎡ (b (r − 1) + C 0 ) (r0 − 1) P000 ⎤ − E [ N v ] = GV′ (1) = ⎢ 0 0 2 ⎥, c0 C 0 (1 − (c 0 + 1) k ) ⎦ ⎣
(45)
⎡b0 (1 − r0 )a 2 b 2 [4 c ′d ′g − 6h(c ′d ′′ + c ′′d ′) − 6e ′(2c ′′′d ′ + 3c ′′d ′′ + 2c ′d ′′′)] ⎣ −2η b0 (r0 − 1)[(2abc ′d ′){ gf + 3hm + 3e ′l }] − [{hf + 2e ′m}(6a ′bc ′d ′ +6ab ′c ′d ′ + 3abc ′′d ′ + 3abc ′d ′′) − 3e ′f (6a ′′bc ′d ′ + 12a ′b ′c ′d ′ + 6a ′bc ′′d ′ E [ N B ] = GB′ (1) =
+6a ′bc ′d ′′ + 6ab ′c ′′d ′′ + 6ab ′c ′d ′′ + 2abc ′′′d ′ + 3abc ′′d ′′ + 2abc ′d ′′′)]] P000 , 24(abc ′d ′) 2 E [ N D ] = GD′ (1) =
α G ′ (1) . β B
(46) (47)
The expected length of working vacation is given by (cf. Yang et al. [32]) E [V ] =
1
η
+
1 , λ0 (1 − BV∗ (η ))
where BV∗ (η ) = SV∗ (η + λ0 − λ0 BV∗ (η )) and SV∗ (η ) = (
(48)
λ0 k ). η + λ0 k
The expected length of cyclic period is given by E [C ] =
E [V ] . PV
(49)
The expected length of busy period is obtained using E [ B ] = E [C ]PB .
(50)
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Similarly, the expected length of brokendown period is given by
E [ D ] = E [C ]PD .
(51)
5. Sensitivity Analysis In order to verify the efficiency of our analytical results, we perform numerical experiment by using MATLAB. The variations of different parameters on average queue length are shown in Figures 1-3. The results of the long run fraction time in different states and expected length of different states for the variation of different parameters are presented in Tables 1. Table 1 investigates the variation of various parameters λ, μ, μ1, α, β, η on the long run fraction time in different states of the server namely (i) PV (ii) PB (iii) PD (iv) the expected length of working vacation period E[V] (v) the expected length of busy period E[B] (vi) the expected length of breakdown period E[D] duration and the expected length of cyclic period E[C]. It is noticed that PV increases as λ, μ1 and β increase but decreases with μ, α, and η. Long run fraction time PB decreases when λ, μ1, α, increase, but increases as μ, β, η increase. Similarly long run fraction time PD decreases with the increase in the values of λ, μ1, β but increases with μ, α, η. We can easily see that expected length E[V], E[B], E[D] and E[C] decrease when λ and μ increase but increase with the increasing value of η. The expected length of working vacation period increases as μ1 and β increase; the expected length of breakdown period increases as α increases, the expected length of busy period increases as β increases. It is noticed that the expected length of cyclic period remains constant with the increasing values of μ1, α and β. It is also noticed that the expected length of busy period and breakdown period decrease with μ1, the expected length of working vacation period and busy period decrease with increasing values of α, the expected length of breakdown period decreases as β increases. Figures 1(a)-(c) depicts the behavior of arrival rate λ0, λ1 and λ2, respectively on the average queue length for k=1, 2 and 3, default arrival rates λ0=0.8, λ1=0.8, λ2=0.7 and fixed parameters μ=2.5, μ1=2, α=0.5, β=5, η=1. As expected the average queue length increases as the arrival rates λ0, λ1, λ2 and k increase however the effect of λ2 is not significant. Figures 2(a)-(c) and Figures 3(a)-(c) show the effect of parameters (μ, μ1,k) and (α, β, η), respectively on the average queue length for homogenous arrival rate (λ0=λ1=λ2=0.5) denoted by continuous graphs and heterogeneous arrival rate (λ0=1.1λ, λ1=λ, λ2=0.9λ) denoted by discrete graph corresponding to k=1, 2, 3. It is observed that the average queue length is higher for heterogeneous arrival rate in comparison to homogenous arrival rate. It is also noticed that average queue length increases as the number of phases of service k increases. From Figures 2(a), 2(b) and 2(c) it is clear that the average queue length increases as μ and k increase but decreases with the increase in μ1. It is seen in Figure 3(a) that E[Ns] increases linearly as α increases but the effect of failure rate α is not much significant. The average queue length E[Ns] decreases as the repair rate (β) and the working vacation rate (η) increase as shown in Figures 3(b) and Figure 3(c). Overall we conclude that •
The average queue length is higher for heterogeneous arrival rate in comparison of homogenous arrival rate.
•
As we expect the average queue lengths, show the increasing and decreasing tends with (λ0, λ1, μ, α, k) and (μ1, β, η) respectively, but the effect of λ2 and α are not
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significant. The effect of k is much significant in comparison to other parameters, which is in agreement with physical situations.
λ 0.100 0.200 0.300 0.400 0.500 0.600 μ 2.000 2.100 2.200 2.300 2.400 2.500 μ1 1.500 1.700 1.900 2.100 2.300 2.500 α 0.100 0.200 0.300 0.400 0.500 0.600 β 1.000 1.500 2.000 2.500 3.000 3.500 η 1.000 1.050 1.100 1.150 1.200 1.250
Table 1. The effect of various parameters on long run time and expected length in different states PB PD E[V] E[B] PV 0.181 0.745 0.074 8.085 33.248 0.184 0.742 0.074 3.766 15.151 0.188 0.738 0.074 1.897 7.436 0.193 0.734 0.073 0.949 3.605 0.199 0.728 0.073 0.414 1.515 0.207 0.721 0.072 0.087 0.303 PV PB PD E[V] E[B] 0.194 0.733 0.073 3.853 14.596 0.192 0.734 0.073 3.542 13.527 0.191 0.735 0.074 3.191 12.277 0.190 0.736 0.074 2.800 10.845 0.189 0.737 0.074 2.368 9.231 0.188 0.738 0.074 1.897 7.436 PB PD E[V] E[B] PV 0.172 0.752 0.075 1.738 7.581 0.180 0.745 0.075 1.815 7.511 0.186 0.740 0.074 1.873 7.458 0.190 0.736 0.074 1.919 7.417 0.194 0.733 0.073 1.955 7.383 0.197 0.730 0.073 1.986 7.356 PV PB PD E[V] E[B] 0.205 0.780 0.016 2.061 7.859 0.200 0.769 0.031 2.018 7.749 0.196 0.758 0.046 1.977 7.642 0.192 0.748 0.060 1.936 7.538 0.188 0.738 0.074 1.897 7.436 0.184 0.728 0.087 1.859 7.337 PV PB PD E[V] E[B] 0.116 0.589 0.295 1.172 5.936 0.143 0.642 0.214 1.446 6.473 0.159 0.673 0.168 1.599 6.782 0.168 0.693 0.139 1.696 6.984 0.175 0.707 0.118 1.762 7.127 0.180 0.718 0.103 1.810 7.234 PV PB PD E[V] E[B] 0.188 0.738 0.074 1.897 7.436 0.176 0.749 0.075 2.519 10.732 0.165 0.759 0.076 3.000 13.802 0.155 0.768 0.077 3.372 16.662 0.147 0.776 0.078 3.660 19.331 0.139 0.783 0.078 3.881 21.822
fraction E[D] 3.325 1.515 0.744 0.361 0.151 0.030 E[D] 1.460 1.353 1.228 1.084 0.923 0.744 E[D] 0.758 0.751 0.746 0.742 0.738 0.736 E[D] 0.157 0.310 0.459 0.603 0.744 0.880 E[D] 2.968 2.158 1.696 1.397 1.188 1.033 E[D] 0.744 1.073 1.380 1.666 1.933 2.182
E[C] 44.658 20.433 10.077 4.914 2.080 0.421 E[C] 19.908 18.422 16.695 14.729 12.523 10.077 E[C] 10.077 10.077 10.077 10.077 10.077 10.077 E[C] 10.077 10.077 10.077 10.077 10.077 10.077 E[C] 10.077 10.077 10.077 10.077 10.077 10.077 E[C] 10.077 14.324 18.181 21.700 24.923 27.885
E[Ns ]
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0.5
0.6
0.7
λ0
0.8
0.9
1
0.8
0.9
1
0.8
0.9
1
(a) 250 k=1 k=2 k=3
E[Ns ]
200 150 100 50 0 0.5
0.6
0.7 λ1
E[Ns ]
(b) k=1 k=2 k=3
200 180 160 140 120 100 80 60 40 20 0 0.5
0.6
0.7
λ2
(c) Figure1. Average queue length vs (a) λ 0 (b) λ 1 (c) λ 2
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k=1 k=2 k=3
k=1 k=2 k=3
2.1
2.2
E[Ns ]
80 60 40 20 0 2
2.3
μ
2.4
2.5
E[Ns ]
(a) 180 160 140 120 100
k=1 k=2 k=3
k=1 k=2 k=3
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(b) 600 λ=.3 λ=.4 λ=.5
500
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300 200 100 0 1
2
3
4 k
5
6
(c) Figure 2. Average queue length vs (a) μ (b) μ 1 (c) k
7
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0.3
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0.5
0.6
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3
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1.15
1.2
1.25
ٛ
(a) 250
k=1 k=2 k=3
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200
k=1 k=2 k=3
150 100 50 0 1
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2 β
E[Ns]
(b) 140 120 100 80 60 40 20 0
k=1 k=2 k=3
1
1.05
k=1 k=2 k=3
1.1 η
(c) Figure 3. Average queue length vs (a) α (b) β (c) η
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6. Conclusion The M/Ek/1 queue with server breakdown, working vacation and state dependent rate has been investigated. The applications of unreliable server model and working vacation in manufacturing process, computer and communication networks are well established. The phase type service and state dependent rates are included in our model which make our model more closer to realistic congestion situations. Several performance indices are derived in explicit form by using generating function approach. Sensitivity analysis has been done which may be helpful to improve the grade of service by selection of appropriate system descriptors. The present investigation can be extended by incorporating bulk input/service. The another direction for future research may be the phase repair of broken down server by including setup time, for which analytical results are not easy to obtain, however numerical technique based on matrix-geometric approach can be implemented.
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15. Madan, K. C. and Al-Rawwash, M. (2005). On the M /G/1 queue with feedback and optional server vacations based on a single vacation policy, Applied Mathematics and Computation, 160(3), 909-919. 16. Madan, K. C. and Al-Rub, A. Z. A. (2004). On a single server queue with optional phase type server vacations based on exhaustive deterministic service and a single vacation policy, Applied Mathematics and Computation, 149(3), 723-734. 17. Medhi, J. (2000). Stochastic Processes. Wiley Eastern Limited, New Delhi. 18. Servi, L. D. and Finn S. G. (2002). M/M/1 queues with working vacations (M/M/1/WV). Performance Evaluation, 50, 41-52. 19. Shogan, A. W. (1979). A single server queue with arrival rates dependent on server breakdown. Naval Research Logistics Quarterly, 26, 487-497. 20. Sztrik J. and Gál T. (1990a). A queueing model for a terminal system subject to breakdowns. Computers and Mathematics with Applications, 19(1), 143-147. 21. Sztrik J. and Gál T. (1990b). A recursive solution of a queueing model for a multiterminal system subject to breakdowns. Performance Evaluation, 11(1), 1-7. 22. Takagi, H. (1990). Time dependent analysis of M/G/1 vacation model with exhaustive service. Queueing Systems, 6(4), 369-389. 23. Takagi, H. (1991). Queueing Analysis. A Foundation of Performance Evaluation. Vacation and Priority Systems, 1, North-Holland, Amsterdam, New York. 24. Takagi, H. (1993a). Queueing Analysis. A Foundation of Performance Evaluation. Finite Systems, 2, North-Holland, Amsterdam, New York. 25. Takagi, H. (1993b). Queueing Analysis. A Foundation of Performance Evaluation. Discrete Time Systems, 3, North-Holland, Amsterdam, New York. 26. Takagi, H. (Ed.), (1993c). Stochastic Analysis of Computer and Communication Systems. North-Holland, Amsterdam, New York. 27. Tang, Y. H. (1997). A single-server M/G/1 queueing system subject to breakdownssome reliability and queueing problems. Microelectron and Reliability, 37(2), 315-321. 28. Tian, N. and Zhang, Z. G. (2006). A threshold vacation policy in multi-server queueing systems. European Journal of Operation Research, 168(1), 153-163. 29. Wang, J. (2004). An M/G/1 queue with second optional service and server breakdowns. Computers and Mathematics with Applications, 47(10-11), 1713-1723. 30. Wartenhorst, P. (1995). N parallel queueing systems with server breakdown and repair. European Journal of Operation Research, 82(2), 302-322. 31. Wu, D. A. and Takagi, H. (2005). M/G/1 queue with multiple working vacations. Performance Evaluation, 63, 654-681. 32. Yang, W. S., Kim, J. D. and Chae, J. C. (2002). Analysis of M/G/1 stochastic clearing systems. Stochastic Analysis and Applications, 20(5), 1083-1100.
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Authors’ Biographies Dr. Madhu Jain, Associate professor of Mathematics, Dr. B. R. Ambedkar University, Agra, received her M.Sc., M.Phil., Ph.D. and D.Sc. degrees in Mathematics from University of Agra. She has been a Gold Medalist of Agra University at M.Phil. level. There are more than 180 research publications in refereed International/National Journals and more than 15 books to her credit. She was recipient of the young scientific award and SERC visiting follow of Department of Science and Technology (India) and career award of University Grants Commission (India). She has successfully completed six sponsored major research projects of Department of Science and Technology (India), University Grants Commission (India) and Council of Scientific and Industrial Research (India). Her current research interest includes Performance Modelling, Soft Computing, Bio-informatics, Reliability, Engineering and Queueing Theory. Twenty candidates have received their Ph.D. degree under her supervision. She has visited more than 25 reputed Universities/Institutes in USA, Canada, UK, Germany, France, Holland, and Belgium. She has participated and presented her research works in more than 30 Internationals and 75 Nationals Conferences/Seminars. Presently she is on the editorial board of several Journals and is referee for the research papers in National & International Journals of repute. Praveen Kumar Agrawal is a research scholar in the Department of Mathematics at Institute of Basic Science, Khandari, Dr. B. R. Ambedkar University, Agra, India. After completing his Master’s degree in first division, he received his M.Phil. degree in Mathematics from Dr. B. R. Ambedkar University, Agra, India. His research interest includes Queueing Theory and Reliability Analysis. He has participated and presented his research work in 7 International and National Conferences/Seminars.
