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44
PERFORMANCE CHARACTERIZATION OF TWO-SERVER BATCH SERVICE QUEUE WITH SECOND
OPTIONAL SERVICE
Andwilile Abrahamu George •
Department of Mathematics and ICT, College of Business Education, Dodoma, Tanzania. gandwilile@gmail.com
P. Vijaya Laxmi •
Department of Applied Mathematics, Andhra University, Visakhapatnam, India.
Abstract
In this paper, we analyze the performance of a finite capacity two-server Markovian batch service queueing model with second optional service. The servers provide two kinds of services, the first essential service (FES), which is provided to all incoming customers and the second optional service (SOS) to those who demand it after completing FES. The service times of the two servers are identical and are exponentially distributed. Matrix-decomposition method is used to obtain the steady-state probabilities of the model. Numerical results and discussion are presented to demonstrate the impact of the model parameters on the system behavior. Furthermore, the cost model optimization is developed to determine the optimal service rates using the Quasi-Newton method to minimize the expected cost. Finally, the findings of this work show that the blocking probability is monotonically decreases as finite buffer size increases and approaches its minimum value of zero when finite buffer is sufficiently large.
Keywords: Two-server, Batch service queue, First essential service, Second optional service.
Mathematical Subject Classification: 60k25, 90B22.
I. Introduction
In everyday life, there are various queueing circumstances where all incoming customers demand the FES and only some may demand the SOS provided by the same server. Currently, this has taken a major consideration by various researchers such as Wang and Kuki [23] wherein they analysed the performance of retrial queueing system with SOS. They obtained the queue length, waiting time and busy period using the method of discrete transformation. SOS with a single server fixed batch service queueing system during repeated vacations has been studied by Ayyappan et al., [3]. They analyzed the model using the probability generating function and Rouche's theorem to obtain the probability of the number of customers present in the queue while the server is busy or on vacation.
Multi-server retrial queue with SOS has been presented by Ke et al., [13]. They derived stationary probabilities using matrix analytic approach. An M/M/1 queue with SOS and working breakdown has been analysed by Yang and Chen [24] who derived the stationary
probability distribution of the system size using the matrix geometric method. Other extensive studies conducted on an assortment of queueing models with SOS are found in Gupur [10]; Kalyanaraman and Murugan [12]; Ke et al., [14]; Uma and Punniyamoorthy [22], etc. For the most part, customers get the service individually. However, this rule may not work in all circumstances, since in some places, the server provide service in batch (groups) of customers instead of serving individually. Batch services are more useful in telecommunication, where data bundle is transmitted in the accumulated large entities (batches), in the field of transportation, in a manufacturing system, in a smart city crowdsourcing application for mobile, etc.
Batch service queue has been broadly studied by numerous researchers. Goswami and Samanta [8] presented the two heterogeneous servers with a discrete-time bulk service queueing system and derived a closed-form expression of the stationary probabilities at arbitrary epoch. The manufacturing bulk service queues using Bayesian hierarchical models are presented by Armero and Conesa [2]. They developed the inferential method for the parameter of the governing model using hierarchical models. Barbhuiya and Guptal [6] analysed the GI/MY/1 queue, wherein the closed form explicit formulations of the system distribution at pre arrival and arbitrary epochs in terms of roots of the characteristic equation generated were derived using the supplementary variable approach and the difference equation method. The queue for bulk service with delayed vacation has been analyzed by Krishna Reddy and Anitha [15]. They obtained stationary probabilities and waiting time distribution of an incoming customer. Ayyappan et al., [4] studied the single server fixed batch service queueing system. The server serves exactly k items at a time. If he finds less than k items in the queue, he takes a multiple vacation of length a. Batch service with multi-server queue model has been presented by Ghimire et al., [7], wherein they computed the stationary probability, queue length size and the amount of waiting time in the queue.
A Markovian single server queueing system has been widely studied by numerous researchers for the last few decades. Due to increasing demand for services, single server operations can lead to congestion in the system such as manufacturing and production, medical clinics, in telecommunication system, etc. Multi-servers have also been studied by various researchers to reduce congestion, Hwang et al., [11]. The queueing system with two servers that are heterogeneous and a variant vacation policy has been studied by Yue and Tian [25], where the two servers take vacations together whenever the system becomes empty. They obtained the explicit expressions of the stationary distribution of the system size. Krishnamoorthy and Sreenivasan [16] investigated a queueing system using two heterogeneous servers, one of which is regularly accessible while the other goes on vacation whenever there are no customers waiting for service and service times of the two servers are exponentially distributed with different service rates. They analyzed the model using matrix geometric method to obtain the mean waiting time in the steady state regime. Kumar et al., [17] investigated the two server queue with heterogeneous servers that were vulnerable to catastrophes wherein any arriving item requires exactly one server for the service and the service rates are different. They obtained both transient and stationary probabilities of number of items in the system using probability generating function and the modified Bessel function of the first kind. Similar studies are found in Ammar [1] and Kumar and Sharma [19] with impatient behavior wherein the service times are independently and exponentially distributed with different service rate. They obtained the explicit transient and steady state probabilities by using probability generating function. Recently, Kumar et al.,[18] generalized the work of Kumar and Sharma [19] by introducing the concept of feedback and reverse balking. They used an iterative approach to obtain the probabilistic measures of the model.
Queues with limited waiting space are more realistic in real-life circumstances. For example, routers servicing incoming packets with varying speeds in a network are examples of this scenario. If the server is busy, the incoming items wait in the queue, but the incoming items are deemed lost when the waiting space is full. One of the major considerations of a system designer in this situation is to provide enough buffer space to keep the blocking probability to a minimum. The limited buffer size queues have been researched by various authors such as Sikdar and Gupta [21],
who analyzed a queue for batch arrival and batch service with a limited buffer size with single and repeated vacations. They obtained stationary probability distributions of the number of items in the queue at departure and pre-arrival epochs. Moreover, they presented the average queue length, waiting time and the blocking probability as the performance measures. Gupta et al.,[9] analyzed a queue for bulk services on a single server with finite buffers and vacation. Using the supplementary variable and embedded Markov chain methods, they obtained the stationary joint distribution of the queue length and the kind of vacation that the server has taken at different epochs. Banerjee [5] presented a queue with a limited capacity and workload-dependent service. They used the embedded Markov chain technique and the supplementary variable approach to obtain stationary probabilities at a departure and arbitrary epoch.
In view of the growing demands for service in practice, our objective here is to investigate the performance of finite buffer two-server batch queue with SOS. In existing literature reviews, no work has been reported with a combination of two-server batch queue with SOS. This inspires us to investigate a two-server batch queue with SOS. Consideration of SOS with varying batch size service makes the model more versatile. The main contributions of this paper are as follows.
• First, we derive the stationary probabilities by using matrix-decomposition method.
• Second, we develop a cost function to optimize service rates for both FES and SOS using Quasi-Newton method so that the expected total cost is minimized.
The remaining part of the paper is structured as follows: Mathematical model description and its formulation are presented in Section 2. In Section 3, we discuss steady state probabilities where the servers are busy or idle in both FES and SOS. The performance measures are discussed in Section 4. The presentation of numerical analysis and discussion is in Section 5. The cost model optimization is developed in Section 6 and in Section 7, we conclude the paper.
II. MoDel DescRipTiON aND Mathematical FoRmulaTiON
We consider a finite buffer M/Mb/2/N queue model with SOS, where arrival occurs according to a Poisson process with parameter A. The servers provide two kinds of service, FES, which is provided to all incoming items and SOS to those whose demand it after completing FES. The item opts the SOS with probability r (0 < r < 1) or leave the system after completion FES with the probability (1 — r). Furthermore, It is assumed that the service times of two servers are independently, identically and exponentially distributed, each with service rate in FES and in SOS. The queue has limited waiting space of size N. The arriving customers are blocked from entering the queue whenever the queue size is N. The average rate of service for FES is given by
= J ^1, for i = 1, = \ 2^, for i = 2.
The average rate of service for SOS is given by
.. = J № for j = 1
\ 2^2, for j = 2, where i and j are the number of servers in FES and SOS, respectively.
The servers process a partial batch up to a maximum capacity size of b. If there are less than b in the queue, one of the servers begins service to those customers. If there are more than 2b waiting in the queue, then the servers take a batch of size b each based on the order of arrival, while others in excess of 2b customers, wait in the queue.
A A A A A
Figure 1: Transition rate diagram for b = 2
Figure 1 is the state transition diagram showing the various transition states of the queueing model.
I. Formulation of Mathematical Model
At time t > 0, let N(t) be the number of items in the queue, S(t) be the number of servers in FES and J(t) be the number of servers in SOS, where S(t) and J(t) are defined as follows:
S(t) = i, 0 < i < 2, and J(t) = j, 0 < j < 2.
We observe that, {(N(t),S(t), J(t)); t > 0} defines a three dimensional Markov process in continuous time with state space
E = {(n,i,j); 0 < n < N; 0 < i < 2; 0 < j < 2, i + j < 2} .
Let the transient probabilities are defined as
Pn,i,j(t) = Pr {N(t) = n, S(t) = i, J(t) = j}, 0 < n < N; 0 < i < 2; 0 < j < 2, i + j < 2,
where P0 i,j(t) is the transient probability that i and j servers are busy with FES or SOS with no customer waiting in the queue, where 0 < i < 2, 0 < j < 2, i + j < 2.
Pn,i,j(t) is the transient probability that all servers are busy with FES or SOS with n customer waiting in the queue, where 0 < i < 2, 0 < j < 2, i + j = 2, 1 < n < N.
The above set of probabilities at steady state are denoted by P0,i,j and Pn,i,j, respectively. The following set of equations are obtained using the probabilistic arguments.
(1 - P0,1,0 + m P0,0,1, APo/o/o + 2(1 - r)^i?0,2,0 + mP0,1,1,
b b
A?0,1,0 + 2(1 - r)m E Pi,2,0 + V2 E PU,1' i=1 i=1
APn-1,2,0 + 2(1 - r)mPn+b,2,0 + V-2Pn+b,1,1, 1 < n < N - b,
APn-i,2,0, N - b + 1 < n < N - 1,
^N-1,2^
(1 - r)^1 P0,1,1 + 2^2P0,0,2 + P0,1,0, P0,1,1,
APn-1,0,2 + Pn,1,1, 1 < n < N - 1,
A Pn-1,0,2 + Pn,1,1,
bb = AP0,0,1 + (1 - r)^ E Pi,1,1 + 2^2 E Pi,0,2 + P0,2,0, i=1 i=1
= APn-1,1,1 + (1 - r)^Pn+bH + 2^2Pn+b,0,2 + Pn,2,0,
1 < n < N - b, = APn-1,1,1 + 2r^1 Pn,2,0, N - b + 1 < n < N - 1, (m + V2 )Pn,1,1 = APn-1,1,1 + 2r^1 Pn,2,0. The steady state equations (1)-(14) can be represented as matrix-form
PQ=0,
where 0 denotes the column vector with all elements equal to zero, and
(-A M12 M13 0 M15 M16
M21 M22 M23 M24 M25 M26 M27
M31 M32 M33 M34 M35 M36 M37
0 M42 M43 M44 M45 M46 M47
M51 M52 M53 M54 M55 M56 M57
M61 M62 M63 M64 M65 M66 M67
M71 M72 M73 M74 M75 M76 M77
is a (3N + 6) x (3N + 6) square matrix. The entries of the matrix Q are listed below:
M12 = (A 0)1x2, M13 = (0 0 ... 0) 1xN, M15 = (0 0 ••• 0)1x 0)
0 2x N
0 A
AP0,0,0 (A + Ц!)P0,!,0
(A + 2^1)P0,2,0
(A + 2m )Pn,2,0 (A + 2^1 )Pn,2,0 2^1 PN,2,0
(A + ^2^0,0,1 (A + 2^2)P0,0,2 (A + 2^2 )Pn,0,2
2^2 PN,0,2
(A + ^ + ^2^0,1,1 (A + ^ + ^2 )Pn,U
(A + m + ^2 )Pn,1,1
Q
1N
M27 M23 =
(0 0 ::: 01n,=(0 0
00 A0
2x N
, M24
2N
2W2x1
■ ^1xN, M21 , M25 =
(1 - r)m 0
21
M16 = (0 0^x2 , M26 = (r01 ^ M31
0
0 0
0 ri2
, M32
N1
00 V0 0/ Nx2
M51
/ a A 0 ■ ■ 0 0
/0\ A10 a A ■ ■ 0 0
0 , M33 = A20 A21 a ■ ■ 0 0
0 Nx 1 an-20 AN-21 AN-22 ■ ■ a A
\AN-10 AN-11 AN-12 ■ ■ AN-1N-2 -2mJ
(1) (2)
(3)
(4)
(5)
(6)
(7)
(8) (9)
(10)
(11)
(12)
(13)
(14)
(15)
2N
NN
where
a = -(A + ), ri2 = 2(1 - r)^, for i = 1,2, ••• , b, and b < N,
A I 2(1 - r)m, if V = 1 for 1 < i < N - 1,0 < j < N - 2, b < N, 11 | 0, otherwise.
0 0
M34
, M35
(2r^1 0 0 2r^1
0
0 0
, M44 = -(A + + ^2),
Nx 1
NxN
M37
00 00
00
0 0
0
\ 0 0 ••• 2r^x)
, M42 = 0)lx2, M43 = (0 0 ••• ^1XN,
NN
M45 = (A 0 ... 0)1x N, M46 = ((1 - r)m rm )1x2, M47 = (0 0 ••• 0)1
1N
M52
0 0 0 ■ ■ 0 0
0 si2 B10 0 0 ■ ■ 0 0
B20 B21 0 ■ ■ 0 0
, M53 =
0 0
0 0 Nx 2 BN-20 BN-21 BN-22 ■ ■ 0 0
\BN-10 BN-11 BN-12 ■ ■ BN-1N-2 0
NN
where
si1 = (1 - r)p1 and si2 = p2, for i = 1, 2, ■ ■ ■ , b, and b < N,
B I m, if V = 1 for 1 < i < N - 1, 0 < j < N - 2, b < N,
ij 0, otherwise.
M55
i A
C10 C20
A A' C21
0 A A'
CN-20 CN-21 CN-22
\Cn- 10 CN- 11 CN- 12
A' A
CN-1N-2 -(H1 + V-2 )J
, M61
21
NN
where A' = -(A + ^1 + ^2).
C.. = J (1 - r)m, if - = 1 for 1 < i < N - 1,0 < j < N - 2, b < N,
ij 0, otherwise.
M22
-(A + ^1) A 2(1 -r)m -(A + 2m)
, M57
22
M62
, M63
22
00 00
(r^1 0 0 rp1
00 , M67
2N
00 A0
0 0
W NxN
... 0
0
2N
M65
M71
0 0 0 0
0 0
0
, M66
2xN
, M72
Nx 1
-(A + V2) 0
2^2 -(A + 2^2)
, M64
2x 2
00 00
00
, M73
N2
00 00
00
2x 1
0
0 0
NN
M74
where
0 0 0 ■ ■ 0 0
/wiA D10 0 0 ■ ■ 0 0
D20 D21 0 ■ ■ 0 0
, M75 =
0
0 Nx 1 DN-20 DN-21 DN-22 ' ■ 0 0
\DN-10 DN-11 DN-12 ' ■ DN-1N-2 0
Nx N
wi1 = 2^2, for i = 1, 2, ■ ■ ■ , b, and b < N,
D = j 2^, if V = 1 for 1 < i < N - 1,0 < j < N - 2, b < N,
0,
otherwise.
M54
/siA
0 0
, M36
M77
Nx 1
/-(A + 2^2) 0 0
00 00
00
, M76
00 00
, M56
N2
00
N2
A0 -(A + 2^2) A 0 (-A + 2^2)
\
00 00
00
0 0 0
N2
-(A + 2) A
0 -2^2/
NN
In the following sequel, we use a matrix decomposition approach to obtain the stationary probabilities in a recursive manner. Let P0 0 0, P1, P2, P011, P3, P4, P5 be the vectors of stationary probabilities where ?i = (P0,i,0, P0,2,0), P2 = (Pi,2,0, P2,2,0, ••• , PN,2,0), P3 = №,1,1, P2,1,1, ■ ■ ■ , PN,1,1), P4 = (P0,0,1, P0,0,2), p5 '= (P1,0,2, P2,0,2,' ■'■ • , Pn,0,2).
From equation (15) it follows that
-AP0,0,0 + P1M21 + P4 M61 P0,0,0 M12 + P1M22 + P2 M32 + P0,1,1 M42 + P3 M52 P1M23 + P2 M33 + P3 M53 P1M24 - P0,1,1 M44 + P3 M54 + P4 M64 + P5 M74 P2 M35 + P0,1,1 M45 + P3 M55 + P5 M75 P1M26 + P0,1,1 M46 + P4 M66 P3 M57 + P4 M67 + P5 M77
(16)
(17)
(18)
(19)
(20) (21) (22)
Equation (21) yields
P4 = P1 + P0,1,1$2,
(23)
where $1 = -M26M661 and $2 = -M46M661. Using equation (23) into equation (16), we obtain
1
Po,o,o = x [?1°3 + po,i,i$4], (24)
where $3 = M21 + $1M61 and $4 = $2M61. From equation (22), we obtain
P5 = P3O5 + Pi$6 + Po,i,i $7, (25)
where $5 = -M^M-1, $6 = -$1M67M771 and $7 = -$2M67M771. Substituting equation (25) into equation (20), we get
P3 = (P2$8 + Po,1,1 $9 + Pi$io)$n , (26)
where $8 = -M35, $9 = -(M45 + $7M75), $10 = -$6M75
and $11 = M55 + $5M75.
Using equation (26) into equation (18), we get
P2 = (P1 $12 + Po,1,1$13 )$-41, (27)
where $12 = -(M23 + $1oM53$-11), $13 = -$9M53$-11
and $14 = M33 + $8M53$-11.
Using equations (27) into equation (26), we have
P3 = P1 $15 + Po,1,1$16, (28)
where $15 = $12$—1 $8$-11 + $10$-1 and $16 = $13$-41 $8$-1 + $9$-11. Substituting equations (24), (27) and (28) into equation (17), we have
Pi = Po,i,i$i7, (29)
and $17 is given by
$17 = - AB-1,
where A = $4M12 + A$13$-41M32 + AM42 + A$16M52,
B = $3M12 + AM22 + A$12 $-41M32 + A$15 M52.
Substituting the value of P^l into equations (23), (24), (25), (27) and (28), respectively, we obtain
P4 = Po,i,i($i7 $1 + $2), (3G) 1
Po,o,o = Po,i,i a ($17 $3 + $4), (31)
P5 = Po,1,i($17 $15 $5 + $16$5 + $17$6 + $7), (32)
P2 = Po,1,l($17 $12 $-4L + $13 $-4L ), (33)
P3 = Po,1,l($17 $15 + $16). (34)
Now all probabilities have been expressed as a function of Po,i,i. The normalization condition is Po,o,o + Po,i,i + Pi ei + P2^2 + P3 e2 + P4ei + P5 e2 = 1, (35)
where ei and e2 are vectors with all of the entries equal to one of dimensions (2 x 1) and (N x 1), respectively.
Substituting equations (29), (3o), (31), (32), (33) and (34) into (35), we get
A >18
Po,i,i = , (36)
$18
where
$18 = $17$3 + $4 + A(1 + $17ei + $17$12$-4 ^2 + $13$-4 g2
+ $17$15 e2 + $16e2 + $17$1 e1 + $2 e2 + $17 $15$5 e2
+ $16$5 e2 + $17 $6e2 + $762)-
The derivation is complete for all stationary probabilities, which can be used to find the measures of performance of the model.
III- Measures of Performance
In this section, we list out some key measures of performance that reflect the bahaviour of the model-
• Let Ls denote the mean number of customers in the system when the servers are busy
b b N
Ls = EE E (n + l + m)P„,i,j, 0 < i < 2, 0 < j < 2, i + j = 2.
1=1m=1n=0
• Let Lq denote the mean number of customers in the queue when the servers are busy
N
Lq = E npn,i,j, 0 < i < 2, 0 < j < 2, i + j = 2. (37)
n=1
• The arriving customers are blocked from entering the queue whenever the queue size is N. In this case the blocking probability (Pb/ocfc ) is given by
Pb/ocfc = PN,2,0 + PN,1,1 + PN,0,2. (38)
• The effective arrival rate is given by
A.eff = A(1 - Pb/ocfc) = A ( E Pn,i,j ) ,0 < i < 2, 0 < j < 2, i + j = 2.
\n=0 J
The expected waiting time in the system using Little's law, we get
Ws = A-.
Aeff
• The expected waiting time in the queue using Little's law, we get
Lq Veff
Wq = f^. (39)
Ae
Percentage Variation (P.V.) in waiting time (Wq) is defined as
P.V. =|(W >" - (W )m1 x 100%,
(Wq )b1
where (Wq)b1 and (Wq)b2 are the waiting time of two values of b.
I. Practical Application of the Model
The queueing model described in this paper has an application in hospital management systems. The model can be applied to situations where the outpatients request an appointment for HIV testing in the clinic centre. The clinic officer monitors the length of booking windows for appointments of the outpatients, but since, in most cases, there is a limited number of doctors, it leads to an unbalanced ratio between the number of outpatients and the doctors. This situation leads to an increase in the length of the booking window and brings the long waiting for appointments. Therefore, to shorten outpatients waiting time, we limit the size of the booking window and assume the clinic centre has a limited slots capacity during a limited length of the booking window. The clinic centre capacity can be divided into sessions/periods of an equal amount of slots. The slots are termed as the maximum size of the outpatient appointments per session. In this scenario, we consider the pooled testing being used to screen the blood of outpatients to detect HIV infections. The slots are taken into service as the primary test for blood sample testing, which is tested in groups. The slots that test positive results opt for a secondary test for further testing. In contrast, all outpatients in the slots are cleared and exempted from further testing if slots test negative. In this situation, appointment, doctors, pooled test (testing in group), limited length of the booking window, primary test and secondary test correspond to the arrivals, servers, batch testing, limited queue capacity, FES and SOS, respectively, in queueing terminology. In the next section, this model application in practice is analyzed with a numerical investigation.
IV. Numerical Investigation
In this section, we present the applicability of the solutions obtained using matrix-decomposition method. We compute the model numerically by taking the model parameters that have close incidence with the practical situations as N = 14, b = 3, A = 0.6, = 0.4, = 0.3, r = 0.4, with the assumption that b < N, where
• A = Appointment rate of outpatients to the clinic centre,
• = Service rate during primary test,
• ^ = Service rate during secondary test,
• r = Probability of opting secondary test,
• b = Maximum number of appointment of outpatients per session,
• N = Maximum capacity of the outpatients at the clinic centre.
Table 1: Variation in different measures of performance with the change in service rate (цl and ^2)
m Lq Wq Pblock m Lq Wq Pblock
0.1 3.745370 6.480820 0.038017 0.1 6.193580 10.32330 0.105915
0.2 1.727790 2.884150 0.001948 0.2 1.423120 2.371980 0.001781
0.3 1.116660 1.861360 0.000308 0.3 0.836311 1.393880 0.000123
0.4 0.836311 1.393880 0.000123 0.4 0.655543 1.092590 0.000029
0.5 0.678974 1.131630 0.000078 0.5 0.569728 0.949556 0.000014
0.6 0.580261 0.967104 0.000061 0.6 0.520022 0.866710 0.000009
0.7 0.513680 0.856134 0.000053 0.7 0.487796 0.812998 0.000008
0.8 0.466371 0.777284 0.000048 0.8 0.465319 0.775536 0.000007
0.9 0.431387 0.718978 0.000045 0.9 0.448812 0.748024 0.000006
1.0 0.404684 0.674474 0.000042 1.0 0.436214 0.727027 0.000005
Table 1 shows the effect of the service rate of primary test and secondary test (^2) on some performance measures. We observe that increasing ) decreases Lq, Wq and Pbhck• This is because as increases, outpatients are served faster so that the queue length and the waiting
time decrease. Moreover, Pbiock tends to zero due to faster services.
0.1K
>
= 0.08
!q <0
o 0.06
CL
0.04
!2 o
2 0.02 m
0
6
•-f.----
-►-b = 1 ■—b = 3 b = 5
__w______
8 10 12 Finite buffer size N
14
Figure 2: Effect of buffer size on the blocking probability
Figure 2 shows the impact of buffer size (N) on the blocking probability (Pbiock) for different batch sizes b. We observe that Pbiock monotonically decreases as N increases and reaches its minimum value zero as N is sufficiently larger. Moreover, when N is kept fixed, Pbiock decreases with the increase of b, as we expect. This is because as b incrases, more number of outpatients are taken for service in a batch, which leads to a decrease in size of the queue. Hence Pbiock decreases.
<1> 6 3 0) 3
<1> 5
£ 4
■H*
O)
Is
n S ■o
£2
o 0) a x
UJ1
-♦-b = 1 ■■-b = 3 b = 5
JT
3
Figure
4 5 6 7 8
Finite buffer size N
3: Effect of finite buffer size on the expected waiting time in the queue (Wq)
In Figure 3, we observe that for fixed N, Wq decreases as b increases. This is because, as b increases, more outpatients are served in a batch at a time, as a result, Wq decreases. Further, for fixed b, except b = 1, the waiting time is more prominently increasing when N lies in [3,6]. However, when the buffer size increases beyond 6, the waiting time is not much affected by increasing N, since the arrival rate is constant and the doctors serve the outpatients in batches b > 1. When b = 1, the waiting time increases monotonically as N increases. The reason is that by increasing the buffer size, more outpatients accumulate in the queue, and the doctor serves one outpatient at a time, this leads to an increase in the queue length. Hence, the waiting time increases compared to b = 3 (b = 5).
3.5
"S 3
2.5
<D
S 2
fl5
<D
3 1
<D
a.
x 0.5
LU
0
N = 6 -N = 10 ►-N = 14
X
J
0.2
0.4
0.6
A
0.8
1.2
Figure 4: Impact of the arrival rate on the expected queue length
Figure 4 shows the impact of A on the expected queue length (Lq) for different values of N. It is clear that as A increases, Lq increases for all values of N, as it should be. Moreover, for a fixed A, as N increases, more outpatients accumulate in the queue thereby an increasing trend can be seen in Lq.
0.15
s 0.1
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en
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o o
CO
N = 6 N = 10 N = 14
Jf
0
0.2
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____................ _ + -
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0.4 0.6 0.8 1
1.2
Arrival rate (A)
Figure 5: Impact of arrival rate on the blocking probability
1
Figure 5 shows the impact of A on Pbiock. As intuitively expected, Pbiock increases with the increase of A. Furthermore, for a fixed A, Pbiock is high for smaller values of N, which is true.
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Case1: P.V. of b = 1 vs b = 3 Case2: P.V. of b = 1 vs b = 5
/
1.2 1.4
0.2 0.4 0.6 0.8 1 Arrival rate (A)
Figure 6: Impact of arrivai rate on the percentage variation in expected waiting time
Figure 6 shows the impact of arrival rate A on the P.V. in Wq in two cases, case 1: P.V. of b = 1 vs b = 3 and case 2: b = 1 vs b = 5. As A increases, the P.V. in Wq for case 1 and case 2 initially increases up to A = 0.7 and for A > 0.7 it slightly decreases. Moreover, the P.V. in Wq varies widely as A increases. This means that as arrival rate increases, there is a high probability of blocking outpatients to enter the clinic centre.
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0 0.1
0.2 0.3 0.4 0.5 0.6 0.7 Arrival rate (A)
Figure 7: Impact of arrivai rate on the anticipated waiting time
For different values of b, we show the impact of A on the anticipated waiting time (Wq) in Figure 7. As A increases, the inflow of outpatients to the clinic centre increases, which tends to a longer queue. Hence, Wq increases. Further, Wq shows an opposite trend with the increase of batch size taken for the service, as intuitively expected.
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r = 0.0 r = 0.5 r = 0.8
0.05
0.1
0.25
0.3
0.15 0.2 Arrival rate (A)
Figure 8: Impact of arrival rate on the anticipated waiting time in the system
Figure 8 demonstrates the impact of A on the anticipated waiting time in the system (Ws) for different values of r. For a fixed r, it is observed that Ws increases with the increase of A, as we expect. Furthermore, for a fixed A, Ws is smaller in the absence of secondary test (r = 0.0), and as r increases, more number of outpatients are tend to secondary test (SOS), which leads to an increase in Ws.
8
56
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0
r = 0.3 r = 0.5 r = 0.7
3
Batch size b
5
Figure 9: Effect of batch size on blocking probability
Figure 9 shows the effect of b on the blocking probability (Pblock) for different r. As b increases, more outpatients are served in batch and leave the queue, which results in a decrease of Pblock. Furthermore, for fixed b, Pblock increases as r increases. The reason is that the outpatients opting for secondary test increase the expected waiting time of other outpatients to be served.
1
0.2 0.4 0.6 0.8
Probability of opting SOS (r)
Figure 10: Effect of probability r on the expected queue length
Figure 10 depicts the effect of r on the anticipated queue length (Lq). We observe that as r increases, outpatients opt for secondary test, thereby increasing the expected queue length. Also, for a fixed r, as batch size increases, more outpatients are served at a time, resulting in a decrease in Lq.
V. Cost Model Optimization
In this section, we present the total anticipated cost function per item per unit time. The main
goal is to figure out the optimal value of service rates y1 and y2 for primary and secondary tests, respectively, so that the cost is minimized. We apply the Quasi-Newton technique to derive the optimal values of the service rates.
I. Quasi-Newton Method
Quasi-Newton method is reliable and efficient for finding a minimizer of a nonlinear function by estimating the curvature along a sequence of search directions with some fixed tolerance (say e). Let yl denote the current point at iteration i = 0, 1, 2, ■ ■ ■ The gradient of f at yl is denoted —Vf(y1 ) and abbreviated to Vf (y). We use H(yl ) to denote a positive definite matrix which is an estimate of the inverse Hessian V2f (yl)-1. It is important to note that if f is not differentiable at y!+1 we say that the algorithm breaks down (in theory) and if Vf (y) = 0 we say it terminates at a smooth stationary point (for more details the reader may refer [?]).
II. Algorithm for Quasi-Newton method
The steps of Quasi-Newton method can be described as follows:
Step 1: Pick any starting trial solution for y0 = (yj, y0) and compute f (yj, y2).
Step 2: While | df | > e or | df | > e; do step 3 - 5.
d2 f d2f ■
ay 2 dy{dy2 d2f df _'ày2ày1 dy2 .
dy1 I I dy2 I
Step 3: Compute the cost gradient Vf (^) = [, ]T and the cost Hessian matrix H
at point V (t), provided that dy: ( \/l ) — ( dyfï )2 = 0, which enables the
existence of inverse of H(p).
Step 4: Update the new trial solution p(i+1) = p(i) - [H(p(i))]-1^f (p(i)).
Step 5: Set i = i + 1 and return to step 2. If the gradient is sufficiently smaller than e, then stop.
To implement the above algorithm, we propose the cost function f (p1, p2) per item per unit time as
f (^1, P2) = CqLq + C1 + 02^2 + 2C3. (40)
Let us consider the following optimization problem:
f (Pi, P2) = Minimizef (p1, ),
s.t
where the various cost components are defined as follows: Cq = cost per unit per item present in the queue, C1 = cost per unit time when the servers are in FES, c2 = cost per unit time when the servers are in SOS, c3 = fixed cost for purchase of one server.
The goal of using this cost model is to give emphasis on the service rates in order to have a cost benefit and less congestion at the queueing system.
We apply the Quasi-Newton technique to obtain the optimal value of service rates (p1, p2). Assuming the values to the coefficients of the expected total cost function in (40) as Cq = $600, c1 = $300, c2 = $400, c3 = $200, e = e1 = e2 = 10-7 and A = 0.6, b = 3, r = 0.4 and N = 14. Starting with the initial values (p0, ^0) = (0.4,0.3), the following Table values have been calculated.
Table 2: Quasi-Newton Method.
Iteration 0 1 2 3
0.69485 0.46377 971.328 0.29561 -2.25317 -3.79388 1098.71 1.79815
_t_j l_j l_j l1.79815 2530.27
Iteration 4 5 6 7
0.69669 0.46515 971.324 0.29376 -0.00029 -0.00050 1088.39 1.78592 1.78592 2497.83
From Table 2, we find that the minimum cost per unit time is f (p2, ) = $971.324 at , ) = (0.69669,0.46515) achieved at seventh iteration.
m m
f P2) Lq
f f
Hessian
0.40000 0.30000 1141.79 0.83631 -914.739 -1275.38 6861.83 2.13400 2.13400 18086.7
0.53331 0.37052 1010.26 0.50344 -308.569 -437.473 2720.75 2.82643 2.82643 6854.12
0.64672 0.43434 974.409 0.34443 -68.0794 -99.3126 1414.49 2.10218 2.10218 3374.53
P2
f P2)
Lq f
f
dF2
Hessian
0.69690 0.46530 971.324 0.29358 0.25677 0.34083 1087.19 1.78470 1.78470 2494.97
0.69667 0.46513 971.324 0.29378 -0.02454 -0.04239 1088.50 1.78605 1.78605 2498.18
0.69669 0.46515 971.324 0.29376 0.00303 0.00402 1088.38 1.78590 1.78590 2497.79
Table 3: The optimal service rates (pi, p2) and cost function f (pi, ^2) obtained in variation ofb and r.
H_H2_Lq f №> H2 )
b = 1 r = 0.2 r = 0.5 r = 0.8 0.80571 0.572062 0.234602 1011.3 0.728169 - - -0.691397 - - -
b = 3 r = 0.2 r = 0.5 r = 0.8 0.687462 0.357727 0.243295 895.306 0.698797 0.505399 0.314466 1000.48 0.700593 0.598757 0.367520 1070.19
b = 5 r = 0.2 r = 0.5 r = 0.8 0.670994 0.327201 0.250341 882.383 0.687876 0.475721 0.321209 989.377 0.697033 0.570477 0.373146 1061.19
b = 7 r = 0.2 r = 0.5 r = 0.8 0.667993 0.319684 0.252881 880.000 0.685428 0.469603 0.323526 987.585 0.695602 0.565179 0.375082 1059.80
b = 9 r = 0.2 r = 0.5 r = 0.8 0.667434 0.317632 0.25369 879.497 0.684913 0.468318 0.32414 987.285 0.695243 0.564205 0.375558 1059.59
In Table 3, we observe that
• For fixed b, as r increases, we observe that pi shows an increasing trend. However, for b = 1, as r increases, the reverse trend is observed in p1. This is necessary in order to balance the system in the profitable manner. On the other hand, it is obvious that p*2, Lq and f (p1, p*2) increase with the increase of r.
Similarly, for a fixed r, as b increases, p1, p*2 and f (pi, p2) all decreases except Lq which increases as b increase the system profitably.
increases as b increase. Lq increase because the service rates decrease in order to balance
• For b = 1 and r = (0.5)0.8, the service rate in secondary test (SOS) does not satisfy the condition p1 > p2. Hence, we exclude those values.
In general, we observe the following features of the queueing system:
• As service rates increases, blocking probability, expected waiting time, and queue size decrease.
• The blocking probability is higher for smaller values of buffer size.
• An increase in arrival rate increases the blocking probability, expected waiting time, and queue size.
• An increase in batch size decreases the blocking probability, expected waiting time, queue size, and optimum cost.
• An increase of r probability increases the optimum cost.
VI. Conclusion
A two-server batch service queueing model with SOS is studied using matrix-decomposition method. The cost model optimization was also developed to determine the optimal service rates to minimize the cost. Performance measures and numerical illustrations discussed in this paper provide valuable insights about the functioning of clinic centre in providing the services
of primary and secondary test. The clinic offices will benefit from the simplified numerical simulations, which will help them increase system efficiency.
In future work, we will incorporate a two-server batch service queue with SOS, adding the concepts of working vacations and vacation interruption. Also, we will consider the transient state in the current model with batch arrival.
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