2017 International Conference on Intelligent Communication and Computational Techniques (ICCT) Manipal University Jaipur, Dec 22-23, 2017
N-Policy for M/Hk/1 System with Unreliable Server and Setup Anamika Jain Department of Mathematics and Statistics, Manipal University Jaipur, Rajasthan - 303 007, India
[email protected]
Praveen Kumar Agrawal
Madhu Jain
Department of Mathematics, GL Bajaj Institute of Engineering and Technology, Mathura-281406, India
[email protected]
Department of Mathematics, IIT Roorkee, Roorkee, Uttarakhand-247 667, India
[email protected]
Abstract: - In this investigation, we study an M/Hk/1 queue with a unreliable changeable server with state dependent rate and setup under N-policy. Arrival of the customers follows Poisson distribution dependent upon the server’s state. The service of the customer starts only when at least N customers are there in the system and turns off when system becomes empty. Queue size distribution is obtained using Generating function method. We derive cost function in term of cost elements related to different situations and determine the optimal operating policy. Sensitivity exploration has been done in order to explore the outcome of dissimilar parameters on cost incurred, threshold level and other indices.
covers the analytical expressions of probability generating functions for different states. In section 4 some important performance indices such as probability of the customers for which the server is in idle, busy, breakdown and in repaired state and the estimated number of customers in the system are determined. The stable condition for steady state has also established. In section 5 we determine the optimal threshold parameter N* which reduces the total expected expenditure of the system. In section 6 numerical demonstrations are given to validate the logical results. In section 7, finally conclusion is elicited.
Keywords:-M/Hk/1 queue, State dependent, N-policy, Unreliable server, Setup, Generating function, Queue size
We made the following assumptions in order to construct the mathematical model: (r, m, j) is the states of the system with r=0, 1, 2,……..; m=0, 1, 2, ……..,k; j=f, w, d, s; where r represents number of customer for the system, m=0 means customer is non-working and m=1, 2,………,k denote that the customer has provision of type m. Also
II. MODEL DESCRIPTION
I. INTRODUCTION In many realistic situations the server takes some time prior to the start service of the customers that is known as the setup or startup time of the server or in other words the setup time is a small amount of time taken by the server before starting the service of the first customer (unit). Operating N-Policy is the act to obtain the best result under the given circumstances. In N-policy server starts service after accumulation of N customers. In N-policy we determine the threshold parameter in order to minimizes the expected total cost. Wang et al. (2005) and Ke (2003) studied the optimal cost of queueing system with server breakdowns and repair using various control policies. Marin and Bulo (2014) studied a queue with hyper exponential service time and Poisson arrival. Yang and Wu (2015) obtained N-policy queue with unreliable server. In this present study we extend the work of Wang et al. (2004) on non-reliable server under N-policy strategy in M/Hk/1 queueing system. By incorporating the setup time and state dependent rate we analyze M/Hk/1 queue with server breakdown under N-policy as follows. We outline some notations and assumptions so as to construct the mathematical classical of the system in section 2. Section 3
978-1-5386-3030-3/17/$31.00 ©2017 IEEE
m
f , S e rv e r is in id le s ta te ° w , S e rv e r is in o p e ra tin g s ta te ° ° ® d , S e rv e r is in b ro k e n d o w n s ta te a n d ° re p a irm a n is in s e tu p s ta te ° s , S e rv e r is u n d e r re p a ir ¯°
The service of customer starts only when N or more customers are there, once the system is empty; however, server turns off only once the system again becomes void. The arrival time of customers is negative exponentially distributes with arrival rates Oj depending upon the server’s state and is given by
138
Oi
° ° ° ® ° ° °¯
W h e n s e r v e r is a n id le a r r iv a l r a te is
O
W h e n s e r v e r is b u s y a r r iv a l r a te is
O1
Expected length of the breakdown (repair) Period is E[D], (E[S])
W h e n s e r v e r is b r o k e d o w n a n d
PI, PB
r e p a ir m e n is in s e tu p s ta te a r r iv a l r a te is
O2
W h e n s e r v e r is u n d e r r e p a ir a r r iv a l r a te is
O3
Fraction of long-run time for which server is idle, and busy respectively.
P D, P S
The service time is hyper-exponential distributed
Fraction of long-run time for which server is breakdown, and under repair, respectively.
with k phases. For busy server state, the customer
The balanced steady state equations are
while waiting for the server is available.
O 50 , f ( 0 )
The server breakdown may take place only when it
O 5 0 , f ( r ),
…(1)
1 d r d N 1
k
is in operational state. The server is sent for
¦
O 50, f (0 )
.. .(2)
P m 5 m , w (1 )
m 1
repairing after breakdown which is rendered by a
k
repair after a random setup time.
( O 1 D P m ) 65 m , w (1 )
q m [ ¦ P j 5 j , w ( 2 )] E 5 m , s (1 ),
m
1 , 2 , 3 ,......., k
j 1
Repair time and life time of the server and setup
.. .(3)
time of repairman are assumed to follow a negative
k
(O1 D P i )5m , w ( r )
exponential distribution.
O 1 5 m , w ( r 1 ) q m [ ¦ P j 5 j , w ( r 1 )] E 5 m , s ( r ), j 1
The customers are served according to FIFO
m
2d r d N 1
1, 2 , 3 ,......., k ;
discipline.
… (4)
The following notations are also used: th
Pm The rate of the m type service,
(O1 D P m )5m , w ( N )
q m O 5 0 , f ( r 1) O 1 5 m , w ( N 1)
(m=1, 2, 3,…….,k)
k
q m [ ¦ P j 5 j , w ( N 1 )] E 5 m , s ( r )
D Server’s failure rate E
j 1
… (5)
Server’s repair rate
qm Probability that the customer enters service of type i,
k
(O1 D P m )5m , w ( r )
(m=1, 2,……..,k)
O 1 5 m , w ( r 1 ) q m [ ¦ P j 5 j , w ( r 1 )] E 5 m , s ( r ) j 1
m
When the server is off - P0,f (r) is the probability
1, 2 , 3 ,......., k ;
r t N 1
... (6)
that there are r (r=0, 1, 2, ………,N-1) customers
( O 2 Q ) 5 m , d (1 )
in the system.
D 5 m , w (1 ),
( O 2 Q ) 5 m , d (1 )
When the server is in busy (working) state and the customer is in service of type m, (m=1,
Q 5 m , d (1 ),
( O 3 E ) 5m , s ( r )
m
...(7)
1 , 2 , 3 ,......... ., k
O 2 5 m , d ( r 1 ) D 5 m , w ( r ),
( O 3 E ) 5 m , s (1 )
2,………..,k) - Pm,w (r), r probable customers in the
m
m
1 , 2 , 3 ,......... ., k ;
r t 2
1 , 2 , 3 ,......... .., k
O 3 5 m , s ( r 1 ) Q 5 m , d ( r ), m
...(8) … (9)
1, 2 , 3 ,......... ., k ; r t 2
system.
… (10)
For server broke down state and the customer is in
III. PPROBABILITY GENERATING FUNCTION
service of type m, (i=1, 2,………..,k) - Pi,d (r) are the probable r customers in the system.
For obtaining the closed form expressions of P0,f (0) we use the generating function technique because recursive method is not enough to find out the expression for P0,f (0). The probability generating functions define as
When the server is under repair state and the customer is in service of type i, (i=1, 2,……..,k) Pi,s(r) is the probability of having r customers in the
N 1
H
system.
0 ,f
(z)
¦ r
n
z 5 0 ,f ( r ) ,
…(11)
z d 1
0
f
Estimated length of the idle (busy) period is E[I],
H
m ,w
(z)
¦ r
(E[B]).
139
0
n
z 5 m ,w ( r ),
m
1 , 2 , 3 ........., k
…(12)
H
¦
(z)
m ,d
n
z 5 m ,d ( r ) ,
m
…(13)
1 , 2 , 3 ........., k
¦
(z)
m ,s
n
z 5 m ,s ( r ) ,
m
Lemma 1: The partial generating functions are expressed as follows:
H
H
m ,s
5 0 , f ( 0 ),
D (z)
m
H
(O 2 z O 2 Q )
DQ H
1 , 2 , ........,
q 2 P 1 H 1 ,w ( z ) q 2 ¦ P jH
…(16)
k
m ,w
(z)
,
m
m ,w
( O 2 z O 2 Q )( O 3 z O 3 E ) q k ¦ P jH
N
),
m
T j (z)
T m (Z )
¦
P0 , f ( 0 )
qm P m T j (z)
ª º DEQ 2 ) z », «O1 z (O1 D P m O O Q O O E ( z )( z ) 2 2 3 3 ¬ ¼
N 1
(z)
(1
¦ r
n
z P0 ,f ( r ) )
0
1 z
(z)
qm ¦ P jH
j, w
(26)
( z ) E zH
k
H (1 )
m,s
N
(z) H
2
j, w
(z)z H
m,w
1 ) 5 0 , f ( 0 ), m
(1 )
summing over r and using (12) & (13), we get
H
m ,w
(1 )
(z)
0
m ,w
(1 ) H
m ,d
(1 ) H
m ,s
(1 ) @
1
…(27)
1
Thus we get 0, f
m ,w
¦ >H
since at z=1 both numerator and denominator are zero. 1 , 2 , ...., k
H
(z) DH
(1 )
apply the L-Hospital rule once in (15)-(18), respectively
(z)
Now multiplying (7)-(8) by relevant power of z and
m ,d
0, f
In order to obtain H0,f (1), Hm,W (1), Hm,D (1) and Hm,S (1) we
…(20)
(O 2 z O 2 Q ) H
H
m
j 1
qm zO ( z
P 0 ,f ( 0 )
The normalizing condition is given by
…(19)
P0 ,f ( r )
k m,w
)
k ª º qm N «QE [ O 1QE O 2 DE O 3 DQ ] ¦ q m U m (QE DE DQ ) » P m 1 m ¬ ¼
evaluate the value of
over n and using (12) & (14), we obtain D P m zH
qm
Pm
Proof: By using normalizing condition and lemma 1 we
1 , 2 , ......, k
N
1 z
m 1
…
Multiplying (2)-(6) by relevant power of z and summing
O1
¦ m 1
Proof: Using (1) and (11), we get
0 ,f
N
q k z O (1 z ) P0 ,f ( 0 )
(QE [ O 1QE O 2 DE O 3 DQ ] ¦
jzi
m
H
(z)
k
k
m 1
m 1
j, w
1, 2 , .......... , k k
(z)
…(25) With the help of Cramer’s rule we solve the equations (23)(25) and obtain the probabilities generating functions. Lemma 2: The probability P 0 , f ( 0 ) , is given by
1, 2 , .......... , k
jzi k
k ,w
j 1
k
T j ( z ) q m O z (1 Z
) z q k P k ]H
k
(z)
where
D (z)
N
q 2 z O (1 z ) P0 ,f ( 0 )
(z)
DEQ
2
[O 1z (O 1 D P k
…(18)
k
j, w
For i=k, repeating this process we get
1, 2 , .......... , k
m
) z q 2 P 2 ]H 2 , w ( z )
j 3
( O 2 z O 2 Q )( O 3 z O 3 E )
N m (z)
( O 2 z O 2 Q )( O 3 z O 3 E )
…(24)
D
(z)
…(23)
k
N m (z)
…(17) H
DEQ
[O 1z (O 1 D P 2
…(15)
5 0 ,f ( r )
1 z
(z)
m ,d
…(14)
Similarly for m=2, equations (20)-(22) give 2
N
1 z
(z) m ,w
N
q 1 z O (1 z ) P0 ,f ( 0 )
0
r
H 0 ,f ( z )
j, w
(z)
j 2
…(14)
1 , 2 , 3 ........., k
) z q 1 P 1 ]H 1 , w ( z )
…(13)
k
q 1 ¦ P jH
f
H
(O 2 z O 2 Q )(O 3 z O 3 E )
0
r
…(12)
DEQ
2
[O 1z (O 1 D P 1
f
…(21)
…(28)
NP 0 , f ( 0 ) N QE q m U m k ª qm º «QE [ O 1QE O 2 DE O 3 DQ ] ¦ » P m 1 m ¼ ¬
5 0 , f ( 0 ), m
1 , 2 ,..... k
…(29)
Again multiplying (9)-(10) by relevant power of z and summing over r and using (13) & (14), we obtain …(22) ( O 3 z O 3 E ) H m ,s ( z ) Q H m ,d ( z ) 0
H
Substituting i=1 in equations (20)-(22) and solving, we obtain
m ,d
(1 )
N DQ q m U m k ª qm º «QE [ O 1QE O 2 DE O 3 DQ ] ¦ » m 1 P m ¼ ¬
5 0 , f ( 0 ), m
1 , 2 ,..... k
…(30)
140
H
m ,s
N DE q m U m
(1 )
f
5 0 , f ( 0 ), m
ª qm º «QE [ O 1QE O 2 DE O 3 DQ ] ¦ » m 1 P m ¼ ¬ k
¦
PD
1, 2 ,..... k
r
z Pm , d ( r )
H
m ,d
(1 )
0
r
DE q m U m
k
¦ m
k ª º qm N «QE [ O 1QE O 2 DE O 3DQ ] ¦ q m U m (QE DE DQ ) » m 1 P m ¬ ¼
1
…(31) where
O
Um
,
Pm
m
1 , 2 , 3 ,.........
…(38)
....... k f
¦
PS
Substituting the value of H0,f (1), Hm,w (1), Hm,d (1) and Hm,s (1) in (27), we obtain the value of probability P 0 , f ( 0 ) .
r
r
z Pm , S ( r )
H
m,S
(1 )
0
DQ q m U m
k
¦ m
1
Lemma 3: The stability condition is k
D(
O2
O3
Q
E
qm
1 O1 ¦ m 1
)
k
…(39)
…(32)
Pm
Theorem 2: Number of expected customer in the system
qm
¦
E[N1] is
Pm
m 1
Proof: The results of finding the stabilized condition 0< P 0 , f ( 0 ) E[S(N*)] < E[S(N*-1)]
N DE q m U m ª
k
O «QD [ O 1QE O 2 DE O 3 DQ ] ¦ ¬
m 1
qm º »
Assuming N as a continuous variable, we obtain the
Pm ¼
approximate optimal value of N* by setting dE [ S ( N )] …(44)
x
The E[S ]
expected E [ C ] PS
length
of
dN
0
,
which provides N*.
repaired
N DQ q m U m k ª q º O « DE [ O 1QE O 2 DE O 3 DQ ] ¦ m » P m 1 m ¼ ¬
VI. NUMERICAL ILLUSTRATION The analytical results established are now illustrated by
…(45)
taking the numerical examples for various combinations of
The following cost elements are used, in order to
system parameters. The parameter corresponding to service
construct cost function: S1
start-up cost when server is in turned-on state
S2
shutdown cost when server is in turned-off state
time b(x)=
(Hk)
distribution k
¦
amOme
Om x
,
¦
am
1, ( x t 0 )
with
p.d.f.
are chosen as k=3,
m 1
S3
Unit time costing for server being on
S4
Unit time costing for server being off
S5
Unit time costing for server breakdown
S6
Unit time costing for server repair
S7
holding unit time costing per customer present in
q1=0.6, q1=0.3 and q3=0.1. Tables 1 depict the effect of traffic intensity UUU forD andrespectivelyon the optimal value N* and the corresponding minimum expected total cost E[S(N*)] for the unlike sets of cost elements and fixing
the system
parameters
The optimized value of control parameter N (say N*) has
O O O O P P P D E Q 1 as given
been determined which minimizes the expected total cost.
below:
142
1
2
Set 1:- S1+S2=200, S3=25, S4= 5, S5=50, S6=5, S7=1
50
Set 2:- S1+S2=200, S3=25, S4= 5, S5=50, S6=5, S7=5
O O O O
45
Set 3:- S1+S2=200, S3=25, S4= 10, S5=50, S6=5, S7=1
O O O O O O O O O O
40
Set 4:- S1+S2=200, S3=50, S4= 5, S5=50, S6=5, S7=1 Set 5:- S1+S2=200, S3=25, S4= 5, S5=100, S6=5, S7=1
O O O O O O
35
Set 6:- S1+S2=400, S3=25, S4= 5, S5=50, S6=5, S7=1
30
E[N 1]
Set 7:- S1+S2=200, S3=25, S4= 5, S5=50, S6=10, S7=1
From Tables 1, it is noted that N* and E[S(N*)] both increase as DEandQincrease for all the sets of costs. We see that the expected total cost decreases when U decreases, but increases as UandU decrease. We also find that the value of optimal parameter N* slightly increases as U decreases, but decreases as U and U decrease. The optimal value and the corresponding minimum expected total cost seem to be more sensitive for holding cost (set 2) and startup/shutdown cost (set 6) in comparison to other sets. In Figs. 1-4 we exhibit the effect of dissimilar parameters on expected queue length E[N1] for the various sets of homogenous and heterogeneous arrival rate, for all these Figures, we observe that the expected queue length is less for homogenous arrival rate in comparison to heterogeneous arrival rate. Also by increasing DEQ and N, the average queue length E[N1] increases as clear from figures 1-4, respectively.
25 20 15 10 5 0
1
1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 D
F ig . 1 : A ve ra g e Q u e u e L e n g t h vs . D
350
O O O O O O O O O O
300
O O O O O O O O O O
E[N 1]
250
VII. CONCLUDING REMARKS In this paper we have investigated the state dependent M/Hk/1 queueing system with server breakdown and setup. The optimal threshold parameter determined can be helpful to minimize the expected total operating cost. The incorporation of state dependent rates and setup time make our model more practical from application point of view. The explicit expressions for expected queue length may be helpful in setting management strategies based on performance indices.
200 150 100 50 0 2
3
4
5
6
7
8
9
E F ig . 2 : A ve ra g e Q u e u e L e n g t h vs . E
143
10
300 O O O O
250
E[N 1]
200
5
{9, 300.35} {9, 287.75} {9, 294.17} {9, 298.31}
6
{12, 423.69} {13, 405.33} {12, 14.66} {12, 420.71}
7
{9, 295.45} {9, 283.15} {9, 289.41} {9, 293.46}
O O O O O O
1
{9, 297.18} {9, 284.78} {9, 291.09} {9, 295.17}
O O O O
2
{4, 545.62} {4, 527.99} {4, 537.03} {4, 542.81}
O O O O O O
3
{9, 299.01} {9, 286.80} {9, 293.01} {9, 297.03}
4 150 100 50
(3, 2)
{9, 305.82} {9, 292.90} {9, 299.48} {9, 303.73}
6
{12, 426.28} {13, 407.77} {12,417.17} {12, 423.28}
7
{9, 297.75} {9, 285.32} {9, 291.65} {9, 295.74}
1
{9, 636.49} {9, 610.55} {9, 623.73} {9, 632.29}
2
{4, 1195.08} {4, 1156.86] 4, 1176.41} {4, 1188.95}
3
{9, 638.46} {9, 612.70} {9, 625.79} {9, 634.28} (3, 3)
4
0 1
2
3
4
5
6
7
8
9
Q F ig . 3 : A ve ra g e Q u e u e L e n g t h vs . Q
{9, 305.82} {9, 292.90} {9, 299.48} {9, 303.73}
5
(9, 645.61} {9, 619.07} {9, 632.56} {9, 641.31}
5
{9, 642.58} {9, 616.23} {9, 629.62} {9, 638.30}
6 7
{13, 923.69} {13, 884.52} 13, 904.39} {13, 917.32} {9, 637.11} {9, 611.11} {9, 624.32} {9, 632.89}
REFERENCES 50
O O O O O O
40
O O O O
[2]
O O O O O O
35 E[N 1]
[1]
O O O O
45
30
[3]
25 20
[4]
15 10
[5]
5 0 1
2
3
4
5
6
7
8
9
N F ig . 4 : A ve ra g e Q u e u e L e n g t h vs . N
Table 1: The threshold parameter N* and corresponding least estimated total cost {N*, E[S(N*)]} UUU Set EQ
1
{8, 144.43} {9, 138.47} {8, 141.51} {8, 143.47}
2
{4, 256.33} {4, 248.23} {4, 252.39} {4, 255.04}
3 4
{8, 146.14} {9, 140.37} {8, 143.31} {8, 145.21} (2, 2)
{8, 152.65} {9, 146.22} {8, 149.51} {8, 151.62}
5
{8, 152.65} {9, 146.22} {8, 149.51} {8, 151.62}
6
(12, 202.87} {12, 194.12} {12, 198.57} {12, 201.45}
7
{8, 145.25} {9, 139.24} {8, 142.31} {8, 144.28}
1
{9, 294.58} {9, 282.34} {9, 288.57} {9, 292.60}
2 3 4
{4, 543.03} {4, 525.55} {4, 534.51} {4, 540.24} (2, 3)
{9, 296.41} {9, 284.36} {9, 290.49} {9, 294.46} {9, 303.23} {9, 290.46} {9, 296.96} {9, 301.17}
144
Ke, J. C. (2003): The optimal control of an M/G/1 queueing system with server vacations, startup and breakdowns, Comput. Indust. Eng., Vol. 44, No. 4, pp. 567-579. Wang, K. H., Wang, T. Y. and Pearn, W. L. (2005): Maximum entropy analysis to the N policy M/G/1 queueing system with server breakdowns and general startup times, Appl. Math. Comput., Vol. 165, No. 1, pp. 45-61. Wang, K.H., Kao, H.T. and Chen, G.(2004): Optimal management of a removable and non-reliable server in an infinite and a finite M/Hk/1 queueing system, QTQM, Vol. 1, No. 2, pp. 325-339. Marin, A. and Bulo, S.R. (2014): Explicit solutions for queues with Hypo- or Hyper-exponential service time distribution and application to product-form approximations, Perf. Evalu., Vol. 81, pp. 1-19. Yang, D.Y. and Wu, C.H. (2015): Cost-minimization analysis of a working vacation queue with N-policy and server breakdowns, Comp. Ind. Engg., Vol. 82, pp. 151-158.
