Научная статья на тему 'ANALYSIS OF TWO VACATION POLICIES UNDER RETRIAL ATTEMPTS, MARKOVIAN ENCOURAGED ARRIVAL QUEUING MODEL'

ANALYSIS OF TWO VACATION POLICIES UNDER RETRIAL ATTEMPTS, MARKOVIAN ENCOURAGED ARRIVAL QUEUING MODEL Текст научной статьи по специальности «Медицинские технологии»

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Ключевые слова
Two vacation / encouraged arrival / reneging / retrial queue / Markov model

Аннотация научной статьи по медицинским технологиям, автор научной работы — Rajeswaran K., Rajendran P., Sanjay K., Shivali S., Ismailkhan E.

In this study,Markovian queuing models, which follow encouraged arrival rates and exponential service rates, are used in a variety of systems, including manufacturing, production, telecommunications, computers, and transportation. Every one has a hectic schedule and little free time in the modern world. Because the customer’s arrival is unpredictable, they cannot complete their task in the allotted time because they cannot predict it. The encouraged arrival, idle server state, busy server state, vacation state, and breakdown and repair state conditions for a single-server Markovian queuing system were all taken into consideration. Vacation time grows acceleratory, and vacation policies abound. This Markovian-encouraged arrival queuing model takes into account customer impatience and retrial efforts to ensure service completion. We calculate the combined probability of these states and compare first-come, first-served with bulk service. The different performance measures have also been explained.

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Текст научной работы на тему «ANALYSIS OF TWO VACATION POLICIES UNDER RETRIAL ATTEMPTS, MARKOVIAN ENCOURAGED ARRIVAL QUEUING MODEL»

RT&A, No 4(80) Volume 19, December, 2024

ANALYSIS OF TWO VACATION POLICIES UNDER RETRIAL ATTEMPTS, MARKOVIAN ENCOURAGED ARRIVAL QUEUING MODEL

Rajeswaran K1,Rajendran P2'* Sanjay K3,Shiyali S4 Ismailkhan E5*

1,3,4BTech student, vellore institute of technology, Vellore, 632014, Tamilnadu, India 2*Professor, vellore institute of technology, Vellore, 632014, Tamilnadu, India 5*Assistant Professor, Panimalar Engineering College,Chennai, 600123, Tamilnadu, India

In this study,Markovian queuing models, which follow encouraged arrival rates and exponential service rates, are used in a variety of systems, including manufacturing, production, telecommunications, computers, and transportation. Every one has a hectic schedule and little free time in the modern world. Because the customer's arrival is unpredictable, they cannot complete their task in the allotted time because they cannot predict it. The encouraged arrival, idle server state, busy server state, vacation state, and breakdown and repair state conditions for a single-server Markovian queuing system were all taken into consideration. Vacation time grows acceleratory, and vacation policies abound. This Markovian-encouraged arrival queuing model takes into account customer impatience and retrial efforts to ensure service completion. We calculate the combined probability of these states and compare first-come, first-served with bulk service. The different performance measures have also been explained.

Keywords: Two vacation, encouraged arrival, reneging, retrial queue, Markov model

We consider the single server to be encouraged for working vacations, breakdowns, and repairs where the server may be down at any time. A Poisson distribution governs customer arrival rates, whereas an exponential distribution governs customer service rates. Arriving customers will join the orbit group if they discover that the server is too busy serving another customer. Kalyanaraman and Sundaramoorthy [1] demonstrated that a Markovian has dependent arrival and breakdown. A single vacation on an unstable bulk server is detailed in Haridass and Arumuganathan [2]. Batch arrivals on an infinite server are investigated in Daw and Pender [3]. Examples of batch queues with inadequate identification are explored in Bar-Lev et al[4]. A generalized bulk queue with the Poisson model is addressed in Neuts [5]. Parveen and Begum

[6] Investigated a general bulk queuing approach with a dual working vacation. In Li and Zhao

[7], a retrial model with a consistent retrial rate, breakdowns, and dissatisfied customers were studied. A dual server queuing model for bulk arrival and service was examined in Kumar and Shinde [8]. Working vacations on bulk arrival queues combined with reneging and interruptions were studied in Vijaya Laxmi and Rajesh [9]. Dual service and two vacations are investigated in the bulk arrival Markovian system by Srivastava et al [10]. They investigated batch arrival and retrial queues using a dual vacation policy and the Markovian queuing model Singh and

[email protected], [email protected]

Abstract

1. Introduction

Srivastava [11]. Som and Seth [12] Explored an M/M/1/N system along with encouraged arrivals. Reduced wait times in an M/M/1/N encouraged customer arrivals, as seen in Khan and Paramasivam [13]. Reneging customers were observed in Som [14] when the M/M/c/N model was used in conjunction with encouraged arrivals.

2. Model elaboration The following presumptions have been taken into account: • The customers follow the First-Come-First-Serve rule

A*(1+w)

w representing

• Customers arrive according to an encouraged arrival with mean

the offered value and accelerated distribution with mean }-.

r

• Customers will enter the retrial queue with probability's'

• After a few tries, the customer notices the accelerated distribution with an average of 1- p and attempts the request from the retrial space.

• When a customer attempts to receive service after a certain amount of time has passed, with probability So

• The server will fail time following the accelerated distribution under the fail rate V\ and from the service to the customer under the repair rate V2.

We have given that the random variable J(t) describes the absolute customers of the system at period't', and R(t) = 0,1,2,3... consider the states, a server is free, a server is busy, the server is on vacation and the server is in breakdown & repair state at period't'.

3. Analysis of system-size distribution From first state 0, j>0 when the server is idle state:

A * (1 + w) Po,o = rP1,o (A * (1 + w) + lfSo) Po, l = tyA, i (A * (1 + w)+ jpSo) Po,j =

From second-state 1, j>o when the -.server is busy state:

(1) (2) (3)

(V1 + Ao + sA * (1 + w) + r) $1,o = (1 - s) p?1,1 + V2$3,o + A * (1 + w)?2,o + (SoPo,1

+A * (1 + w)Po,o (1 - Ao + sA * (1 + w) + lr + V1) P1, i

(4)

V2P3, l + rP2, l + A * (1 + w) Po,i + sA * (1 + w) P1, i-1

+ (1 - s)(l + 1)pP1, l+1 + (l + 1)pSoPo, l+1 (1 - Ao + V1 + jr + (1 - s)jp + sA * (1 + w)) PX] (5)

rP2>j + V2Py + A * (1 + w) Po,j + sA * (1 + w) P31,-1

+sA * (1 + w) (j + 1) pP1,j+1 + (j + 1) pSoPo,j+1

(6)

From third state 2, when the server is on vacation:

2A * (1 + w) P2,0 = A0JP0,1

(7)

(p + A * (1 + w)) P2, l = (1 - Ao) Pi, l + A * (1 + w) P1, l_i VP2,i = (1 - Ao) P1,j + A * (1 + w) P1,h1

For fourth state 3, when the- server breakdown and is repaired state:

(8) (9)

V2 P3,0 = Vi Pi,o V2 P3,1 = Vi Pi, I V2 P3,j = Vi Pij

The following conclusions may be drawn from equations (i), (2), and (3).

(10) (11) (12)

A,0 = f^^) Pofi

Pi,m = ( A*(1+r ^) p0, l

p = fMi+W+M^ p 11'j = l ju J 10'j

Using (4), now with j = 0 and (10), we obtain

(13)

(V1 + A0 + s" * (1 + w) + u) i51,0 = (1 - s) fPu + V1 p1,0 + A * (1 + w)p2,0 + f¿op0,1

+A * (1 + w)p0,0 (A0 + s" * (1 + w) + u) Pl1,0 = (1 - s) ^P1'1

+A * (1 + w)p2,0 + ç$0p'0,1 + A * (1 + w)p0,0

If we use (1), we obtain

/ A0 A * (1 + w) + s (A * (1 + w))2

V V

When k=1, use (13)

p0,0 = (1 - s)<pp1,1 + A * (1 + w)p2,0 + Ç00150,1

p^ = (A*(i+w)+^°) p>0,1

p1,1 = «1 p0,1

A A0A*(1+w)+s(A*(1+w))2Nj p^ o

(1 - s) f «1130,1 + A * (1 + w)p2,0 + Ç00pp0,1

If we use (7)

A0A * (1 + w) + s(A * (1 + w))2 ¥

P0,0

(1 - s + «1 + ¿0) f + y

p0,1

PU

2 (A0A * (1 + w) + s(A * (1 + w)A)2)

(1 - s + «1 + ¿0 ) f + f

. . b2 , p0,0 P0,1 = — p0,0 C2

(14)

(15)

M

Rajeswaran K, Rajendran P, Sanjay K, Shivali S, Ismailkhan E

ANALYSIS OF TWO VACATION POLICIES UNDER RETRIAL ATTEMPTS, RT&A, N° 4(80) MARKOVIAN ENCOURAGED ARRIVAL QUEUING MODEL_V°lume 19, December, 2024

Applying j=1 in (6)

(1 - Ao + V1 + r + (1 - s)p + s" * (1 + w)) Pw = pP2,1 + V2J53,1 + A * (1 + w)Po,1

+s" * (1 + w)P^ + 2s" * (1 + w)pPx,2 + 2pSojPo,2 To do this, use (12) and assign j=1.

(1 - Ao + V1 + (1 - s) p + sA (1 + w) + r) P1,1 =rP^2,1 + V1 Pw + A (1 + w) i3o,1 + sA (1 + w) P1,o +2s1 A (1 + w) Px,2 + 2pSoPo,2 [1 - Ao + (1 - s) p + r + sA (1 + w)] Pu = rP2,1 + A (1 + w) Po,1 + s1,oP1,o + 2sxA (1 + w) Pu + 2pSoPo,2

If we use (9) for j = 1 and (14), we get

rP2,1 = (1 - Ao) P14 + A (1 + w) P1,o ((1 - s) p + r + s" (1 + w)) «1 jPo,1 = A (1 + w) Px,o + A (1 + w) Po,1 + s" (1 + w) Px,o +2s" (1 + w) SoP1,2 + 2pSoPo,2 ((1 - s)p «i + r«i + s«i - A (1 + w)) P3o,i = A (1 + w) P1,o + s" (1 + w) P1,o

+2s" So Pu + 2 pSo i3o,2 (16)

Now, we use (13) put j = 2, and we obtain

Pu = ( A (1 + w>r + 2pSo ) i-o,2 = ^ i-o,2

Additionally, using (1) and (15), we can solve (16) to obtain

Po,2 = -3 i'o,o (17)

b3

C3

s" (1 + w) «1 b2 + p (1 - s) «1 b2 +r«1 b2 - A (1 + w) b2 C2

-(A (1 + w))2(1 + s)

r

C3 = 2s" (1 + w) So b1/c1 + 2Sop When you enter j=2 in equations 6, 9, and 12, we get

(1 - Ao + V1 + 2r + 2 (1 - s) p + s" (1 + w)) P12

= rP2,2 + V2J^3,2 + A (1 + w) Po,2 +sA (1 + w) P1,1 + 3s" (1 + w) pP1,3 + 3pSoi5o,3 ... (18)

rP2,2 = (1 - Ao) P1,2 + A (1 + w) P1,1 (19)

V2 P3,2 = V1 i'1,2 (2o)

Rajeswaran K, Rajendran P, Sanjay K, Shivali S, Ismailkhan E

ANALYSIS OF TWO VACATION POLICIES UNDER RETRIAL ATTEMPTS, RT&A, No 4(80) MARKOVIAN ENCOURAGED ARRIVAL QUEUING MODEL_V°lume 19 December, 2024

If we use (19) & (20) in (18), we have

(2 * p + 2 (1 - s) < + s" (1 + w)) Pi,2 = A (1 + w) (1 + s) P51,1 + A (1 + w) Po,2

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+3s" (1 + w) <p?1,3 + 3<pSoPo,3 (21)

(2 * p + 2 (1 - s) < + s" (1 + w)) —Po,2

= A (1 + w) (1 + s) ?1,1 + A (1 + w) P0,2 +3s" (1 + w) <f31,3 + 3<SoJ5o,3

^(2 * p + 2(1 - s)< + s" (1 + w))b1 - A (1 + wfj Po,2

= A (1 + w) (1 + s)?1,1 + 3s" (1 + w) <?1,3 + 3 föoPo,3

(2 * p + 2(1 - s)< + s" (1 + w))b1 - A (1 + w)) -Po,o

c1 ) c3

= A (1 + w) (1 + s)«1 i3o,1 + 3s" (1 + w) <?1,3 + 3<pöoPo,3

^(2 * p + 2(1 - s)< + s" (1 + w))- A (1 + w)^ PPo,o

b2 . . = A (1 + w) (1 + s)«1—fo,o + 3s" (1 + w) <?13 + 3<SoPo,3 c2

/{(2 * p + 2 (1 - s) < + s" (1 + w)) | - A (1 + w)} |\ ]$oo V -A (1 + w)(1 + s^g )

= 3s" (1 + w) <P1,3 + 3q>5oPo,3 If, we Put j = 3 in (13), and we have

P1,3 = (A(1+w)+j'So) ]Po,3 = IPo,3 2 * p + 2(1 - s)< + s" (1 + w))| - A (1 + w)} ^

b2

-A (1 + w) (1 + s)«i — c2

= (3s'bf + 3<¿0) J3o,3 _ (2 * p + 2(1 - s)< + s" (1 + w))b1 - A (1 + w)} | - A (1 + w)(1 + s)«1 |) P

Po,3 ^ (3A(1+w)< b4 +3<,o) i0,0

Po,3 = ^ Poo

In general, we get If we use (13), we get

If, we use (9)

Po,n = Po,1 + Po,2 + Po,3 + ■ ■ ■ (22)

(A (1 + w) + j^oN

Pj = (A(1 + w)+ j"So) (üo,1 + üo,2 + Po,3 + -)

(23)

P2„ = (1 - A0 ) ( iû+wj )

( - - ) A (1 + W) P\ : 1

(P0,1 +1">0'2 + 1">0'3 + •••) + —-(24)

i1

Similarly, if, we use (12), we get

p3,j = (V^) (P0'1 + 110,2 +110,3 + ■ ■ ■) (25)

Now, if we use equations (22), (23), (24), and (25), we obtain

Prn = P0,1 + 1,2 + P0,3 + ■ ■ fors = 0 i A (1 + W )+ (P01 + P0,2 + p0,3 + ■ ■ ■) ,

fors = 1 (1 - A0 ) ( ^jj ) (H0,1 + ^ +10,3 + . . .) + A (1 + W) j ,

V j *u ) u

fors = 2, ^(Pu + P02 + p0,3 + ■ ■ ■) ,fors = 3.

To calculate the value of 110,0, the normalization function, we have

3 œ

EE Pis = 1

s=0 j=0

(p + P + P ) fl + A (1 + w)+ jf¿0 + M

(P0,1 + P0,2 + P0,3 M 1 +--:-2--1--

V j * U2 V2 J

+A (1 + w)

F

-Pl,j-1 =1

Poo(b-2 + ^ + h) (1 + A(1 + ") + j+ Ü ' \C2 C3 C5J \ j * F 2 V2

+A (1 + p -i

—F— j =1

1 a(1+w) pp. p = _1 F l1'}~1_

0,0 fk + b3 + (1 + A(l+^)+j'¿0 + nA

^2 + C3 + c5) y1 + j*F2 +

4. Validation of the model

(i) When the server is free:

Po = j PPo,j

1 A(1+^) pp. . 1 F l1'}~1

(ii) When the server is busy:

( 1 + A(1+^)+j,¿0 + ( h + k +

V1 + j*F2 + V2) \ C2 + C3 + c5)

[Pp0,1 + P0,2 + PP0,3 + ' ' ']

P1 = j P51,j

1 A(1+w) p . _ 1 p P1,j-1

f. + A(l+w)+;'¿o + vA fh + b3 i b5 A V1 + /V + v2j Vc2 + C3 + C5J

(A(1 +■ ) X [JP0,1 + Po,2 + + •••]

V / * F )

(iii) When the server is on vacation:

P2 = Ey=0 P2,/

1 A(1+œ) pp. . _ 1 F Pl,j-1

( 1 + A(1+œ)+f^0 + vA f b2 + b3 + b5 A V1 + j*F2 + + C3 + csj

(1 - Ao )

A (1 + œ) + j'ôo

/ * F2

(jPo,1 + ¿0,2 + P0,3 + ' ' ') +

A (1 + œ) f51//-1

F

(iv) When the server is in a breakdown and repaired state:

P3 = j P3,j

1 _ A(1+w) ■

F

V1

(1 + + (I +1 + D Vv2

X (]Po,1 + Po,2 + Po,3 + ■ ■ ■)

X

5. Conclusion

The Markovian Encouraged Arrival Queuing Model has been developed with the inclusion of customer retry efforts, balking, and reneging behavior. The four system states idle state, busy state, vacation state, breakdown state, and repair state have all been taken into consideration utilizing the concept of encouraged arrival. We have examined and verified the possibilities of the various conditions. Neural networks, communication systems, post offices, and supermarkets can all benefit from using this model to reduce the reneging and balking behavior of their customers.

References

[1] Kalyanaraman, R., & Sundaramoorthy, A. (2o19) A Markovian Working Vacation Queue with Server State Dependent Arrival Rate and with Partial Breakdown. International Journal of Recent Technology and Engineering. 7(6s2), 664-668.

[2] Haridass M., & Arumuganathan R. (2oo8) Analysis of a Bulk Queue with Unreliable Server and Single Vacation. International Journal Open Problems Computational Mathematics. 1(2), 37-55.

[3] Daw, A., & Pender, J. (2o19) On the Distributions of Infinite Server Queues with Batch Arrivals. Queueing Systems, 91, 367-4 1.

[4] Bar-Lev, S.K., Parlar, M., Perry, D., Stadje, W., & Van Der Duyn Schouten, F.A. (2 7) Applications of Bulk Queues to Group Testing Models with Incomplete Identification. European journal of operational research 183, 226-237.

[5] Neuts, M.F. (1967) A General Class of Bulk Queue with Poisson Input. Annals of Mathematical Statistics 38(3), 759-77 .

[6] Parveen, M.J, & Begum, M.I.A. (2013) General Bulk Service Queueing System with Multiple Working Vacation. International Journal of Mathematics Trends and Technology 4(9), 163-173.

[7] Li, H., & Zhao, Y.Q. (2005) A Retrial Queue with a Constant Retrial Rate, Server Break Downs and Impatient Customers. Stochastic Models 21(2-3), 531-550.

[8] Kumar, J., & Shinde, V. (2018) Performance Evaluation Bulk Arrival and Bulk Service with Multi Server Using Queue Model. International Journal of Research in Advent Technology. 6, 3069-3076.

[9] Vijaya Laxmi, P., & Rajesh P,(2018) Variant Working Vacations on Batch Arrival Queue with Reneging and Server Breakdowns. East African Scholars Multidisciplinary Bulletin. 1(1), 7-20.

[10] Srivastava, R.K., Singh, S., & Singh, (2020) A Bulk Arrival Markovian Queueing System with Two Types of Services and Multiple Vacations. International Journal of Mathematics and Computer Research. 8(8), 2130-2136.

[11] Singh, S., & R.K. Srivastava, (2021) Markovian Queueing System for Bulk Arrival and Retrial Attempts with Multiple Vacation Policy. International Journal of Mathematics Trends and Technology ■.

[12] Som, B.K., & Seth, S., (2017) An M/M/1/N queuing system with encouraged arrivals.Global journal and pure applied mathematics vol 13,no7.

[13] Khan, I. E., & Paramasivam, R. (2022). Reduction in Waiting Time in an M/M/1/N Encouraged Arrival Queue with Feedback, Balking and Maintaining of Reneged Customers. Symmetry,14(8), 1743.

[14] Som, B.K., (2018) M/M/c/N queuing systems with encouraged arrivals, reneging, retention, and Feedback customers. Yugoslav journal of operation research, 28(3), 333-344

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