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M/M(A,B)/1 MULTIPLE WORKING VACATIONS QUEUING SYSTEM WITH HETEROGENOUS ENCOURAGED ARRIVAL
PRAKATI.P*, JULIA ROSE MARY.K
Department of Mathematics, Nirmala College for Women, India *[email protected] [email protected]
Abstract
The concept of Queuing system is most commonly used in our everyday life. It is essential to characterize the practical queuing characteristics in order to improve the performance of the queuing model. This study investigates M/M(a,b)/1/MWV queuing model with heterogeneous encouraged arrival occurring in the regular busy period. The considered model follows General bulk service rule and if the system is not in use, or when it is vacant, the server goes on vacation, thus there occurs multiple working vacations which are exponentially distributed. In this study, a model of multiple working vacation queues in which with heterogeneous encouraged arrivals following Poisson process is examined. With the mentioned conditions, the explicit formulations for the steady state probabilities and the performance measures of the proposed model is derived. Also,some particular cases have been developed and compared with existing models. Finally, the numerical impact of various parameters on performance attributes are also analysed.
Keywords: Heterogeneous Encouraged Arrival, General Bulk Service Rule, Multiple Working
Vacations (MWV), Poisson Distribution, Mean Queue Length
1. Introduction
Several fields including telecommunications, traffic signals, the medical sector, inventory and control etc., widely use queuing theory. In a vacations queuing system, there may be a chance of unavailability of a server from a primary service center for a period of time. A vacation is a period of time spent away from the main service location and it can be caused by a variety of circumstances. Neuts'[7] general bulk service rule (GBSR), states that the server processes the clients in batches. Many Researchers have contributed their findings in Queuing theory with server vacations.
Initially, a general type of bulk queues is discussed by Neuts, who also examines the length of the queue and its busiest times. Further,Y. Levy and U. Yechiali[4] discussed about the vacations in queuing model. A class of semi-vacations policies were first presented by L.D Servi and S.G Finn[9], in which the servers operate at a reduced pace rather than suspending all primary service altogether while on vacations and such vacations is termed as Working Vacations.
De-An Wu and Hideaki Takagi[1] examined M/G/1 queue with multiple working vacations and evaluated its performance measures. Liu et al.,[5] stated the queue size probabilities of M/M/1 Multiple Working Vacation Queuing model. Further, K. Julia Rose Mary and M.I. Afthab Begum[3] analysed a single server with bulk service queue with general arrival pattern following multiple working vacations period. Later, K. Santhi and S. Pazhani Bala Murugan[8] examined on the concept of heterogeneous service on M/G/1 queue with two-stage service under single
working vacation. Numerous Researchers analysed the concept of multiple working vacations. Further, O.C. Ibe, O.A. Isijola [2] also discussed on M/M/1 multiple vacations queuing systems with differentiated vacations.
While B.K. Som, S. Seth [10] evaluated M/M/1/N Queuing system with Encouraged Arrivals. Recently, S. Malik, R. Gupta [6] analysed the Finite Capacity Queuing System with Multiple Vacations and Encouraged customers. This paper discusses the concept of heterogeneous arrival specifically encouraged arrival in the busy period state of M/M(a,b)/1/MWV and obtained with performance measures.Also, Prakati and Julia Rose Mary[?] discussed the concept of encouraged arrival with both single and multiple working vacations performing with single server. Further, the numerical analysis of the considered model is evaluated.
2. Methods
An M/M(a,b)/1 queuing model of multiple working vacations with heterogeneous encouraged arrival is considered with the following assumptions:
The three different states of arrival is assumed to be heterogeneous and are uniquely denoted. The heterogeneous arrival process in this model is considered to follow Poisson distribution with parameter Aw( in the idle state , \wv in the vacation state, whereas the heterogeneous encouraged arrival process also follows Poisson distribution with parameter Xwb(1 + 5). According to Neuts (1967) general bulk service rule (GBSR), the server processes the clients in batches. This rule states that the server will only begin providing service if at least "a" customers is present. After completing a service, if the server discovers "a" (or) more clients but not more than "b" clients in the system, he serves them all at once; if he discovers more than b, he serves the first b-customers in turn while the others wait.
As a result, there are minimum of "a" units and maximum of "b" units in each batch for service. The assumption is that the service time of batches of size s(a < s < b) is an independent random variable with identical distribution and a parameter with an exponential distribution "^w". When a service is finished and there are less than 'a' clients in the queue, the server starts vacations, which is an exponentially distributed random variable with parameter q' .If the system length is still less than "a" after finishing one vacations, the server takes another vacations, and so on, until the server detects at least "a" customers in the queue (i.e., multiple vacations is used).
During vacations, if the queue size reaches at least "a" customers, then the server begins providing service at a service rate "^wv" which is different from the regular service rate. The size of the batch in service is "k" with a < k < b and the service rates are independent of the size of the batch in service, thus when the vacations is over the server will shift the service rate from ^wv to when the server is operating. In this model, there is an increase in the arrival rate, i.e., encouraged arrival occurs in the regular busy period and then the server still continues to serve following GBSR. The above queuing model is denoted as M/M(a,b)/1/MWV with heterogeneous encouraged arrival.
3. Steady State Equations Let NK(t) = number of customers waiting at the time, "t"
and M(t) = 0,1 or 2 denotes that the server is idle during the vacations or working during vacations or in the regular busy period respectively.
Let IRn (t) = Pr{Nk(t) = n, M(t) = 0} ; 0 < n < a - 1
VQn (t) = Pr{Nk(t) = n, M(t) = 1}; n > 0
BPn(t) = Pr{Nk(t) = n,M(t) = 2}; n > 0
when M(t) = 0, the size of the queue and the system are same,
when M(t) = 1 or 2, then the size of the system is the sum of total no. of customers waiting in the queue or the size of the service batch containing a < k < b customers.
The Steady State Probabilities satisfying the Chapman Kolmogrov equations are assumed as
follows:
VQn = A wVQn (t);
IRn = A *>IRn (t);
BPn = A ™BPn (t) For the specified model, the steady state equations are expressed below:
XwiIRo = UwBPo + VwvVQ0 (1)
KiIRn = KiIRn-1 + UwBPn + UwvVQn ; 1 < n < a - 1 (2)
b
(Awv + n + Uwv )VQo = XwiIRa —1 + ftwv
E VQ (3)
n=a
(Awv + n + Uwv )VQn = AwvVQ
n—1 + ftwv VQn+b ; n > 1 (4)
b
(Awb(1 + 5) + Uw)BPo = Uw E BPn + nVQo (5)
n=a
(Awb (1 + 5) + Uw )BPn = (Awb (1 + 5))BPn-1 + U
wBPn+b + nVQn) ; n > 1 (6)
In the above mentioned steady state equations, Eq (1) and Eq (2) represent idle state, working vacations are defined in Eq (3) and Eq (4). Furthermore, encouraged arrival occurs during the regular busy period i.e., in Eq (5) & Eq (6)
3.1. Steady State Solution
The concept of forward shifting operator (E) is introduced on BPn and VQn to solve the above defined steady state equations,
E(BPn) = BPn+1; E(VQn) = VQn+1; for(n > 0)
The homogeneous difference equation is obtained from Eq (4)
(UwvE(b + 1) - (Xwi + n + Uwv)E + Awi)VQn = 0;n > 0 (7)
The characteristic equation of the difference equation is expressed as follows h(w) = (Uwvw(b + 1) - (Awi + n + Uwv)w + Awi) = 0.
By assuming f (w) = (Awi + f] + Uwv)w and g(w) = Uwvw(b + 1) + Awi, it is known that if |g(w)| < |f (w)| on |w| = 1, then by Rouche's theorem, h(w) has only one root zv inside the contour |w| . As the root is real, solution of the homogeneous difference equation is obtained as
VQn = znvVQ0) n > 0 (8)
Similarly, Eq(6) can be written as
[UwE(b + 1) - (Awb(1 + 5) + Uw)E + Awb(1 + 5)]BPn = -f]z^VQ0; n > 0 (9)
Again by Rouche's Theorem, Eq (9) is obtained as
[pww(b+1) - (Awb(1 + 5) + pw)w + Awb(1 + 5)] = 0 has a unique root 'z' with |z| < 1 provided < 1
The non-homogeneous difference equation is solved and the solution obtained is given by
where
Y
BPn = (Xzn + Yznv )VQo
i] Zv
-; ifzv = z
(10)
(11)
(Awb(1 + 5))(zv - 1) + UwZv(1 - Zvb) by adding Eqs (1) & (2) over 0 to n, and substituting VQn & BPn , IRn is obtained as follows
IRn = [-P
yw fX(1 - zn+1) , Y(1 - zvn+ )\ , ywv (1 - zvn+1)
+
+
] VQo
Awi\ (1 - z) (1 - Zv) J Awi (1 - Zv)
Hence, the steady state queue size probabilities are expressed in terms of the unknowns X and VQ0.
Now to calculate X , consider Eq (5) and on substituting the value of BPn, it is found that
(za—zb+1) (za—z b+1)
X(Awb(1 + S) + yw) - yw^—^ = n - Y((Awb(1 + S) + yw) - yw^—) (12)
(1 - z)
which can be simplified as
Xpw (1 - za) _
(1 - zv)
n
Yyw (1 - zva )
(13)
(1 - z) VV(1 - zv)J (1 - zv) Moreover, Eq (3) is also verified and the steady state queue size probabilities are expressed in
terms of VQo and obtained as
VQn = zvnVQo ; n > 0
(14)
BPn = (Xzn + Yzvn)VQo ; n > 0
(15)
IRn = ïA± (X^z+l + YOp^) + ^ ^^]VQo; 0 'AwiK (1 - z) (1 - zv) J Awi (1 - zv)
where
- z) (1 - zv)
X- (1 - z)
yw (1 - za ) \\(1 - zv )
n
- zv)
Yyw (1 - zva) (1 - zv)
The value for VQ0 is evaluated by using the normalizing condition E«=0(VQn+ BPn) + En=QIRn = 1
< n < a - 1 (16)
(17)
t (zvn)VQ0 + (Xzn + Yzn)VQ0 + E [At {^y^T1 + n^TTT")
n=0 n=0 Awi (1 - z) (1 - zv)
ywv (1 - zvn+1)]
Awi ( 1 zv)
] VQ0 = 1
Thus (VQ0 1) = F(zv, Vwv) + XF(z, pw) + YF(z v, ) ; where F(r, t) = (1 + £(a - 3-?))
4. Performance Measures
One of the objectives of this paper is to deduce the expected queue length of the considered model. The expected queue length is also known as Mean Queue length and denoted as Lq
4.1. Mean Queue Length
The expected queue length Lq is calculated as
<x a—1
Lq = E n(VQn + BPn) + E nIRn (18)
n=1 n=0
by substituting the values of VQn, BPn and IRn , Lq is simplified as
Lq = XH(z, Uw) + YH(zv, Uw) + H(z vr Uwv )
where
H( )= r t (a(a — 1) ara+1 (1 — r) — r2 (1 — ra
H(Trt) = JY—r)2 + Aw(1 — r) \+ (I—)2
as X and Y are given by Eq (15) & Eq (16)
Also,the other characteristics of the queuing model includes if Pv, Pbusy and Pidie respectively which denotes the probability that the server is in working vacations state, in regular and is idle in vacations stater then
Pv = E (VQn) = ^Q0T (19)
n=0 (1 — zv)
Pbusy =E-=o (BPn) =((1^ + (T—) VQ0 (20)
Pbusy = () + j—j) VQo rwhere X & Y are specified in Eq (17) & Eq (11)
a—1
Pidle = E !Rn (21)
n=0
where IRn is given by Eq (16). Thus, the above characteristics helps in deciding the efficiency of the queuing model.
5. Particular Cases
5.1. Case 1: M/M/1 Multiple Working Vacations Model
The M/M/1 Multiple working vacations queuing model's steady state queue size probabilities are calculated.
By letting in equations Eq (14) to Eq (16) at a= b = 1 and Awb (1 + 5) = Aw, it is observed that
VQn = znvVQo ;n > 0 BPn = y (znv+1 — znv+1) ; n > 0
zv
IRn = where 'z' = Awi = p, ; n > 0
zv U
where
X =- YP and Y- nz
Vw (1 - zv) + (zv - p)
z
v
Prakati.P, Julia Rose Mary.K RT&A, No 4(80)
M/M(A,B)/1 HETEROGENEOUS ENCOURAGED ARRIVAL Volume 19, December, 2024
which results in the queue size probabilities of M/M/1 Multiple Working Vacations Queuing model (Liu et al.2007)
5.2. Case 2: M/M/(a,b)/1 Multiple Working Vacations Model
Letting \wb(1 + S) = Aw,it is obtained as M/M(a, b)/1 under working vacations.
VQn = znvVQo ;n > 0 BPn = (Xzn + Yzn)VQo ; n > 0
IRn
Pw ( X(1 - zn+1 ) + Y(1 - zn+1 ) + Vwi (1 - zn+1 )
Awi( (1 - z) (1 - zv) Awi (1 - zv)
VQ0;0 < n < a - 1
By substituting the values of VQn, BPn and IRn in (Lq) , it is observed that (Lq)= X H(z,pw) + YH(zv, pw ) + H(zv, p-wv)
a(a-1) ! asa+1 (1-s)-s2(1-sa) '
As H (s, t)= + -^r^ as (1-s)-
v ; (1—s)2 ' Awi(l-s) y 2 (1-s)
where X = (1-z) ff-J— ^ YPw (l-zva ) where X = pw^(1-za) { ^ (1-zv) J (1-zv)
Y = _'lis_ if z = z
Awi(zv-1)+Pwzv (1-zvb) v
which coincides with the queue size probabilities of M/M(a,b)/1 Multiple Working Vacations (K.Julia Rose Mary,2011)
6. Sensitivity Analysis
The numerical values of the mean queue length are derived for M/M(a, b)/1 MWV queuing system with heterogeneous encouraged arrival during busy period. The obtained numerical values are examined to study the impact of the vacation parameter rj and also various other parameters like encouraged arrival rate Awb(1 + 5), regular service rate (uw), service rate during vacation (uwv).
Based on the above parameters, the mean queue length (Lq) i.e., the expected system size of batch arrival has been calculated. Further, a comparative study is made between Lq of M/M(a, b)/1 MWV Encouraged Arrival and Lq of M/M(a, b)/1 MWV under Heterogeneous Encouraged arrival.
Thus the impact of the mentioned parameters on the system size probabilities and on the expected system size during the idle state, working vacation state and the regular busy period are also analysed by considering zv and z as the roots of the following characteristic equations
(Uwvw(b+1') — (Awv + rj + Uwv )w + Awv ) = 0 &
(Uww(b+1) — (Awb (1 + 5) + Uw )w + Awb (1 + 5) = 0
respectively.By considering homogeneous arrival rate during idle and vacation state as Aw = 4.05 and during encouraged arrival Aw (1 + 5) = 5.051, also, by assuming for heterogeneous arrival queuing system arrival rate as Awi = 3.9; Awv = 4.0& during encouraged arrival Awb(1 + 5) = 5.051 and further by assuming Uw = 0.9 as constant and by varying rj & Uwv, the mean queue length (Lq) is calculated and tabulated below.
Table 1: Lq ofM/M(a, b) /1 MWV Queuing system with Heterogeneous EA and M/M(a, b) /1 MWV Queuing system under Homogeneous EA
ywv n Lq (Heterogeneous EA ) Lq (Homogeneous EA)
0.02 211.4142 211.5021
0.04 159.4418 159.5500
0.05 0.06 89.8687 89.9613
0.08 64.5690 64.6545
0.1 48.6767 48.7549
0.02 127.0639 127.1303
0.04 105.3322 105.4188
0.1 0.06 74.3123 74.3973
0.08 52.9088 52.9864
0.1 46.01627 46.0943
0.02 100.1150 100.1765
0.04 83.3706 83.44902
0.15 0.06 64.1551 64.2356
0.08 45.1962 45.2689
0.1 41.5132 41.5887
0.02 75.1551 75.2117
0.04 72.1117 72.1873
0.2 0.06 52.6958 52.7703
0.08 41.8486 41.9209
0.1 39.3368 39.41266
From the above tabulation, it is evident that the queue length of the heterogeneous encouraged arrival is comparatively less than that of the homogeneous encouraged arrival, as the arrival rate differs in each state of the heterogeneous encouraged arrival model. The obtained numerical values are represented graphically.
Figure 1: M/M(a, b) /1 MWVwith Heterogeneous Encouraged Arrival
Figure 2: M/M(a, b) /1 MWV with HomogeneouEncouraged Arrival
The above graphical representations demonstrate that the mean queue length reaches its peak at the minimum values of fi&pwv and the mean queue length reduces at a great margin level when t]&pwv are maximum. For example if pwv = 0.05 , and fj= 0.02 , the mean queue length of M/M(a,b)/1 MWV with heterogeneous encouraged arrival is 211.4142 while the mean queue length of M/M(a,b)/1 MWV queuing model with encouraged arrival is 211.5021. Thus with the varied arrival rates there arises a gradual variation in the queue length.
As discussed earlier, the encouraged arrival is an impact factor in deciding the queue length. Thus on varying the encouraged arrival rate during the regular busy period as Awb (1 + 5) = 5.051 ; Awb (1 + 5) = 5.053 ; Awb (1 + 5) = 5.055 and by having Awi = 3.9, Awv = 4.0 &pw = 0.9 as constant and on varying pwv from 0.05 to 0.2 and fj from 0.02 to 0.1 , the mean queue length is computed and tabulated below
Table 2: Lq of M/M(a, b) /1 MWV Heterogeneous Queuing system with varied EA
ywv n Lq (Awb (1 + S) = 5.051) Lq (Awb (1 + S) = 5.053) Lq (Awb (1 + S) = 5.053)
0.02 127.0639 127.06636 127.0687
0.04 105.3322 105.3360 105.3399
0.1 0.06 74.3123 74.3164 74.3205
0.08 52.9088 52.91281 2.968
0.1 46.01627 46.02055 46.0248
0.02 100.1150 100.1170 100.11900
0.04 83.3706 83.37881 83.37708
0.15 0.06 64.1551 64.1588 64.1624
0.08 45.1962 45.1998 45.2034
0.1 41.5132 41.5172 41.5212
0.02 75.1551 75.1566 75.1582
0.04 72.1117 72.1146 72.1174
0.2 0.06 52.6958 52.6990 52.7022
0.08 41.8486 41.8520 41.85550
0.1 39.3368 39.3407 39.3446
Thus from the above tabulations, it is clear that as the encouraged arrival rate increases gradually from \wt (1 + 5)= 5.051 to \wt (1 + 5) = 5.055 the queue length increases gradually from 211.4142705 to 211.4180277. Similarly, when the encouraged arrival rate increases from Kb(1 + 5)= 5.051 to Awb(1 + 5)= 5.055 the queue length increases from 211.4142705 to 211.4217896 . Thus it is clear that as the encouraged arrival rate increases,the queue length increase. Hence the encouraged arrival rate significantly decides the queue length of the considered model. The above tabulations is represented graphically below
700 600
500
| 400
Lq
300 200 100 0
^—5.051 ^—5.053 ^—5.055
I___I
Figure 3: M/M(a, b)/1 MWV Heterogeneous Queuig system with varied Encouraged Arrival
Thus for the vacation service rate pwv varying from 0.05 to 0.2, with the increased encouraged arrival rate, the corresponding mean queue length is denoted in the above graph.
Now, the other performance measures of the discussed model like Pidie. Pvacation, ?tusy are computed by using Eq (19), Eq (20) & Eq (21) and tabulated below:
Table 3: Other Characteristics of M/M(a, b)/1 MWV Heterogeneous Queuing system under encouraged arrival
Uwv n Pidle Pvacation Pbusy
0.02 0.01438 0.7468 0.2387
0.04 0.02275 0.6294 0.3477
0.1 0.06 0.0319 0.5935 0.3745
0.08 0.04156 0.5786 0.37973
0.1 0.04845 0.54449 0.4070
0.02 0.0170 0.7791 0.2038
0.04 0.0261 0.6685 0.3053
0.15 0.06 0.0352 0.6168 0.3478
0.08 0.04602 0.6010 0.3529
0.1 0.05232 0.5584 0.3892
0.02 0.02104 0.8105 0.1683
0.04 0.02933 0.6895 0.2811
0.2 0.06 0.0400 0.6451 0.3147
0.08 0.0496 0.61013 0.3402
0.1 0.05560 0.5644 0.3799
From the above table, as the working vacation service rate (pwv) increases and with the vacation parameter varying from 0.02 to 0.1, the probability values during the idle state and busy period increases and the probability during the vacation period decreases.
7. Conclusion
M/M(a,b)/1 MWV queue under heterogeneous encouraged arrival of customers is studied with steady state and the steady state solutions are derived. Additionally, the mean queue length and various other performance measures were also calculated and numerically examined. Moreover, other queuing models were deduced as particular cases.
From the numerical examples computed for the mean queue length of the considered model and compared with M/M(a,b)/1/MWV with encouraged arrival, evidently concluded that many factors like vacation parameter, service rate plays a role in deciding the queue length, the key factor is the encouraged arrival rate that makes a notable impact in deciding the queue length of the considered batch service heterogeneous queuing model with single server.
Conflict of Interest: The authors declare no conflict of interest
8. Acknowledgement
The corresponding author would like to acknowledge the UGC Scholarship team for approving and providing (SJSGC) Scholarship Grants, which is more likely helpful and beneficial to complete this study.
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