ANALYSIS OF ENCOURAGED ARRIVAL MULTIPLE WORKING VACATION QUEUING MODEL UNDER THE STEADY STATE CONDITION Текст научной статьи по специальности «Медицинские технологии»

ANALYSIS OF ENCOURAGED ARRIVAL MULTIPLE WORKING VACATION QUEUING MODEL UNDER THE STEADY STATE CONDITION

Ismailkhan Enayathulla Khan, Rajendran Paramasivam*

^Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology, Vellore, India - 632014.

[email protected] Correspondence email: [email protected]

Abstract

Businesses typically entice customers with alluring offers and discounts. Encouraged arrivals is the name given to these curious clients. In certain situations, the service offered by queuing models, notably in transportation networks, enables the simultaneous serving of several consumers. In general, closed-form solutions to bulk service queuing models with idle servers are difficult to find. By coordinating the operations at each workstation using the Chapman-Kolmogorov research technique, the main objective of this study is to assess the performance of the car assembly line in order to reduce waiting times. The server is in a busy state, is idle, is regularly busy, and is in a busy state when it breaks down. Performance metrics are being tracked using a multiple working vacation approach. In this study, analysis of encouraged arrival multiple working vacation queuing model under the steady state condition. In this model, we included encouraged arrival. By resolving difference equations and Chapman Kolmogorov balancing equations, the steady state queue size problem is found. Additionally, the server is in the busy, idle state, regular, and breakdown busy states, and performance metrics are conducted. The server was sent for repair and is now completely repaired to avoid the crash at any time. After that, the server continues to offer the service. It is evidently identified that the efficiency level increased while the encouraged arrival is incorporated. The main contribution of this paper is to show the server is in the busy, idle state, regular, and breakdown busy states, and performance metrics efficient level increases. It is found that they offer more efficient results when compared with the Poisson process method

Keywords: Encouraged arrival, steady state, multiple working vacations, single server, queue size.

I. Introduction

There are times when the service provided in queuing models, particularly in transportation systems, allows for the simultaneous serving of many customers. In general, bulk service queuing models with idle servers are challenging to solve in closed form in [1]. A generic class of bulk queues with encouraged input was researched in [12].

A study on the examination of a GI/M/I- queue with several vacations is described in [2]. The Markovian M/ (q, P)/1 queuing model while taking several working vacations was also examined in [4]. An M/M/1 lines with working vacations (M/M/1/WV) model was investigated in [15]. A well-considered time-dependent bulk queuing service solution issues with queuing in [3].

Analyses the best management strategy for a heterogeneous M/M/1- queue with server downtime [5]. An analysis of the N-Policy and the M/M/1 queue with numerous working vacations is found in [10]. The queuing procedure using bulk services is examined in [19]. A Study on the Analysis of the M/G/1 Queue's Queue Length Distribution with Working Vacations [6].

Stochastic models with matrix geometric solutions were examined in [18]. GI/Geo/1 queue

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ANALYSIS OF ENCOURAGED ARRIVAL MULTIPLE WORKING VACATION Volume 18, September 2023

with several Vacations were investigated in [9]. Working vacation queue and matrix analysis a

study was done in [7]. The optimal operation of a Markovian queuing system with a transportable

and unreliable server was examined in [14].

The M/M/1-queue with a single working vacation was investigated in [21]. The M/M/1 queue with a single working holiday and setup times were investigated in [22]. A matrix analytical approach for the examination of the M/G/1 queue with exponential working vacations was looked at in [8 and 16]. Recent advancements in the queuing and bulk models were examined in [17]. An M/M/1/N queue system with encouraged arrivals was investigated in [23].

A finite and infinite M/H2/1 queuing system with a detachable, unreliable server was examined in [20]. The M/M(a,b)/1/MWV/Br Model [25] was investigated. Research on an M/G/1 queue with numerous working vacations may be found in [11]. Reducing wait times in an M/M/1/N encouraged arrival line by providing feedback, balking at unpaid customers, and sustaining those customers was investigated in [24]. A break-down-prone portable service station with M/Ek/1 queuing system optimization was examined in [13].

II. Model Recitation

In this model recitation provided by

• This method is encouraged with parameters A* (1+§). In the manner of General-Bulk Service-Rule (G-B-S-R), the server handles the customers in batches.

• As a result of this rule, the server only begins to provide service when at least a "q" customer is present.

• The server serves the first p - customers, leaving the others in the line.

• Each batch of units must have a certain minimum and maximum number of "q" units to be used.

• The assumption is that the batch size a(q<a<p) will have an accelerated distribution of the parameter and will be an independent random variable with an identical distribution.

• When the server breaks down with the parameter (1 - e_M).

• If the queue- size reaches minimum "q" imagine a situation where a server starts offering service while on vacation at a different rate from the standard one.

• The server was sent for repair and is now completely repaired to avoid the crash at any time. After that, the server continues to offer the service.

Let KA(t) represent the number of customers waiting in line at time t, and C(t) represent one of zero, one, two, or three depending on whether the server is idle, busy, in a typical busy state, or experiencing a breakdown respectively. Let Mk(t) = Prob{KA(t) = k,C(t) = 0}; 0 < k < q - 1, Ak(t) = Prob{KA(t) = k, C(t) = 1};k > 0, Pk(t) = Prob{KA(t) = k, C(t) = 2};k > 0, Yk(t) = Prob{KA(t) = k, c(t) = 3};k > 0, The queue-size and the system-size are equal for C(t) = 0.

Mfc = lim Mfc(t)Mfc = lim Qfc(t);Pfc = lim Pfc(t);Yfc = lim Yfc(t);

t.^ro t.^ro t.^ro t.^ro

As an outcome, the Chapman-Kolmogorov equations that satisfy the condition are as follows:

A* (1+S)Mo= M.Po+^A) (1)

A* (1+8) Mfc=A* (1+fl)Mfc_i+pPfc+^4fc;(1<fc<q-1), (2)

(A* (1 + 8)+/ + ^) A) =A* (1+8) (3)

(A* (1 + 8)+/ + ^) =A* (1+8) ¿fc+jS;(fc > 1) (4)

(A* (1 + fl)+p + s) Po ^^Pfc+zAo + fcYo, (5)

(A * (1 + -8) + p + s)Pfc = A * (1 + -8)Pfc_i + ^P^ + + ¿Yfc; (ft > 1), (6)

(A* (1 + fl) + 6)Yo = sPo, (7)

(A * (1 + 8) + 6)Yfc = sPfc + A * (1 +8) ££=iYfc_n (ft > 1). (8)

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The forward shifting operator Expectation on Pk and Ak are introduced as follows and will be used

to solve the steady-state equations:

Exp(Pk) = Pk+1;Exp(Ak) = AK+1;Exp(Yk) = Yk+1; (k > 0).

Therefore, the homogeneous differential equation is given by equation (4).

(X* (1 + в) + x + Vu)Ak = X* (1+ Шк-i + VuAk+p; (k > 1),

[циЕхрР+1 — E xp (X * (1+в)+Х + ^и) + Х* (1 + в))]Лк = 0;(k> 0), (9)

The difference characteristic equation is provided by

g(o) = [nuoP+1 — (X* (1 + в) + x + Vu)o + K1 + в)] = 0

by taking e(o) = (X* (1 + в) + x + VJo; h(o) = цио^+1 +X* (1 + в)

It is found that lh (o)l < le (o)l on lol = 1.

The solution of the homogeneous differential equation is given by,

= mkuA0 ; (k>0) (10)

Additionally, equation (6) will be expressed as, (X* (1 + в)+ц + s)Pk = X(1 + d)pk-i + цРк+p + XAk + bYk; (k > 1),

[цЕхрР+1 — Exp(X * (1+в)+ц + Б)+Х* (1+ в)]Рк = —xAk+1 — bExp(Yk); (k > 0). (11)

By applying, Rouche's theorem, we discover that the equation,

— (X* (1 + в) + ц + s)o + X * (1 + в) = 0 has unique root with Iml < 1provided by ^+^<1.

Equation (8) will be expressed as follows:

(X* (1 + в) + b)Yk =sPk+X* (1+ в)Гк-1; (к > 1)

Y _ _sExp(Pk)__. .

k (X* (1+d) + b)Exp—X* (1+fl) . ( )

By substituting (11) to (12), We obtain,

[,Exp^+1 — Exp(X * (1 + в) + ^ + s)+X* (1 + фк = —Xrni+1A0 — b [Xt tJ+^X (1+J (13) Consequently, the non-homogeneous difference equation (13) has the following solution:

V

Qmk--^- - Л

,mi+1 — (X* (1 + в)+, + 5)ши + Х* (1+в) + (X* {1 + Q)+Sb%u—X, {1 + ву

(i.e)Pk = (Qmk+Y mku). (14)

The sequence Mk(0 < к < q — 1) for the condition equation (1) & (2) adding, we get

X* (1 + в)£кп=0Мк=Х* (1 + в)£кп-=10Мы+^км=0Рм+^=0Ап, к к

X* (1 + в) Mk=^PN+Vu^

Л

rN + V-u ^ nN.

n=0 N=0

It is discovered by the interchange Ak and Pk in equations (9) and (14) that

^iDMo'

X* (1+в) Mk = [M(Qmk + Ymk) + ^и(тк)]А0,

Mk =

fi (Q(1 — mk+1)Y(1—mku+1)\ fiu (1—тк+1

(1 + в)( (1—m) 1 — mu J X* (1+в)( 1—m

A* I \ (1 ill.) 1 mu

Since Q and Ao are unknowns, the steady-state queue size probabilities are also unknown. To determine Q, we now take into account the equation (5),

P

(X* (l + d)+^ + s)Po= V^Pk+XAo + bYo

k=q

Equations (14) and (12)'s Pk will be substituted to determine that

(X* (1 + 9) + M + S)(Q + Y;M,, = ,(QP-^]+Y

1—mu

)A0+xA0 +

b(-^-

V(X* (1+d) + b)mu—X* (1+d))

It is expressed as,

4 P+i т.. —m,.

и

(1 —m) (1 —mu) (1—mu) V(À * (1 + ö) + è)(£xp — À * (1 + ö))

((À*

^(1 —m1^ x Y/z(l-mU) , u( _ spi

(1

f>(1-m«) s&m l-^ X s&Ymu

_ l>(1-mg)__s&m 1 1 r_

Ç [ (1-m) À* (1+fl)(m-1) + fcmJ [(:

(1_m) X* (1+fl)(m_1) + fcmJ [(1_mu) (1_mu) X* (1+fl)(mu_1) + fcmu As an outcome, the steady-state queue-size probabilities i4o and are provided by, h =

= m^; (ft > 0), (15)

Pfc = (Çmfc + YmU)^o; (ft > 0), (16)

|>(1-m^) s&m l-^ X Y^(l-m£) s&Ymu

_ p(1-m<)__sfom 1 1 [

Ç L (1-m) À* (1+fl)(m-1)+YmJ [

(1-m) À* (1+fl)(m-1)+YmJ |_(1-mu) (1-mu) l«(1+S)(mu-1)+6m„ Where,

y _ _xmu(A(1+i9) (m„-1)+im„)_

[à* (1+0) (mK-1)+^mu(l-m^)+smu][l» (1+fl)(mu-1) + fcmu]-sfcmJ

Mfc = (.d-m!:1) + Yd-mlH)) + (1-m^)l *(o < < q -i). (17)

* Là* (1+0) V. (1-m) (1-mu) / à* (1+0) V 1-m„ /J v ■> \ !

Using the normalizing condition, the expression for is determined as follows:

Hfc^O ^fc + Hfc^O ^fc + Hfc =0 ^fc + Hfc^O Yfc = i-Which follows that,

= £(mu,^u) + Ç£'(m,^) + Y£(mu,^) + Ç5(m,s) + YS(mu,s). 1 N 5 f J o/ c-N 1 r 1

Where fCj,5) = —1— [l +---— (<? — and S(jr,<5) = —L , , ,

v J (1-rc) L À*(1+fl)V (1-re) /J v J (1-re) Là* (1+fl)(re-1) + fcreJ

Thus, the value .-1 is evaluated.

III. Evaluating Performance

In this section, we have performance metrics of M/(q, P)/1/MWV/Ym computed. The expected queue- length (ia) is,

ia = zro=1 ft(^fc + ^ + Yfc) + tm ftMfc. (18)

By Substituting the values of i4fc , Pfc, Yfc and Mfc from (15) to (17), we get

la = tro-1 ft K + (<2rnfc + Ym£)) + YT\ ft + + +

a i->k-1 V u w u/y |_x* (1_m) (1_m„) / (X* (1+fl) V (1_m„) /J

k

yrn j^i_smQm"____smuYmg_\

^fc=1 ((À* (1+fl) + fc)m-A(1+tf) (À* (1+fl) + fc)m„-A(1+tf)/

V(X* (1+fl) + fc)m_A(1+fl) (X* (1+fl) + fc)mu_A(1+0)>

Moreover, a can be simplified as,

ia = + YG(mu,^) + G(mu,^u) + QC(m,s) + YC(mu,s). (19)

Where G(7t,5)=-^ + -^- f^U ^+1C1-*)_f(1_*2)},

v 7 (1_^)2 A(1+0)(1_rc) I 2 (1_^)2 J

C^ 5) = (1_jr)2 [^(1+i)(7r_1)+fc^].

Now Pu, Pfcy (¿usy), Pje (ide) and P^ are given by,

p _ vœ —.

ru = ^fc=o.fc =

(1-mu)

= X Pfc = Z(Çmfc + Y™UMo -fc=0 fc=0 = S:0Mfc,

ç Y

.0,

(1 — m) (1 — mu). p = yœ Y = _^_[ m 1 .__Ysmu_[" mu 1

k (à* (1+fl) + fc)m-À* (1+fl) [(1-m)2J (À* (1+fl) + &)mu-À* (1+fl) [(1-mu)2J.

IV. Conclusion

The analysis of encouraged arrival multiple working vacation queuing model under the steady state condition is discussed in this study in the context of the server being busy, idle state, regular, breakdown busy-state. Using Chapman Kolmogorov balancing equations, the total probability generating function was determined for this model. This model is more effective than the comparative Poisson arrival model [25].In the future to be included EASTA property for this encouraged arrival multiple vacation queuing model.

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