ANALYSIS OF MARKOVIAN BATCH SERVICE QUEUE WITH FEEDBACK AND SECOND OPTIONAL SERVICE Текст научной статьи по специальности «Математика»

ANALYSIS OF MARKOVIAN BATCH SERVICE QUEUE WITH FEEDBACK AND SECOND OPTIONAL SERVICE

P. Vijaya Laxmi*,1/ Hasan A.B.D. Qrewi2 and Andwilile A. George3

Department of Applied Mathematics, Andhra University, Visakhapatnam, India.

Email: [email protected], [email protected], [email protected] * Corresponding author.

Abstract

The aim of this paper is to analyze a single server batch service queue model with feedback and second optional service under a transient and steady state environment. The server provides the first essential service (FES) to all customers who arrive at the system and the second optional service (SOS) to those who need it. After completion of FES, if the customer is not satisfied with the service, he may rejoin the queue or may opt for SOS or exit the system with a particular probability. The service times of both FES and SOS follow exponential distribution. We use the probability generating function and the Laplace transform expression to obtain probabilities in the transient state after inverting Laplace transforms into the time domain. Also, we apply the Tauberian property in the Laplace transform expression to get the steady state probabilities. Finally, some performance measures and numerical results are provided.

Keywords: Batch service queue; Transient state; Feedback; First essential service; Second optional service.

I. Introduction

In queue theory, the customers are served one by one or served in batches whose sizes are fixed or variable in size. When the customers are served, they will depart, but when the customer does not seem satisfied they will return to the queue again to be served; this situation is called feedback. Queueing model with feedback have been studied by many researchers such as [1] who is the first to introduce the concept of feedback mechanism in queues which includes the probability of the customer to back the counter and take the service. Later, [2] investigated a single server with feedback wherein the queue is formed in two categories, one is formed in a waiting room and the other is formed in the service room and obtained the queue size, waiting time, and total time spent in the system. [3] presented a single server model with limiting behavior of the waiting time process and having a certain feedback property. Lemoine [3] classified the queue into two types: the primary queue where customers receive a maximum of time units of service. The secondary queue is formed from the customers who are not satisfied with the service in the primary queue. Those customers are attended to only when all customers who enter the system before them have departed and when the primary waiting room is empty.

[4] discussed M/G/1/ro queueing system with instantaneous Bernoulli feedback. They obtained the time-stationary distribution of the number of customers in the system at an arbitrary epoch. The queueing system with delayed feedback has been studied by [5]. They considered two servers; a lower server with a general service and an upper server with an exponential service. The decision to feedback or not depends on the queue length at the two servers. Other existing works on feedback are found in [7], [10], [9], [11], [12], [14], [15], [16], [17], etc.

In many real service systems, customers want both essential and optional services provided by the server. More precisely, we may consider a system where a single server's service is classified

into two phases. The first phase is required for all customers and only some of them are routed to the second phase service. [6] studied a single server batch arrival and arbitrary service time distribution queue system with second multi-optional service and a finite number of immediate Bernoulli feedback. They provided a steady state analysis of the model, including the asymptotic behavior under a high rate of retrials.

[8] studied the priority retrial queue with immediate Bernoulli feedback and second multioptional service. In this case, the customers' feedback after completing both phases of services. They use the embedded Markov chain technique and probability generating function to obtain the queue size and orbit size. [13] and [17] presented the queue system with batch arrival and multi-optional service with feedback. They consider two kinds of independent customers' arrival: positive and negative arrival. Positive arrivals in batches with the Poisson process and negative arrived singly with the Poisson process. The arrival of a negative customer removes the positive customer in service from the system and makes down the server.

The batch service queues have potential applications in various areas, including manufacturing, production, computer networks, cargo loading and unloading at a harbor, etc. In this situation, the number of items is processed in batches with a limit on the number of items taken at a time for processing. A number of researchers have made substantial contributions to batch service models. [19] investigated a batch service retrial system with feedback in a steady state. They considered two kinds of the arrival of the customers; the positive customer who is served in batch and the negative customer who arrives in the system and removes the positive customer. Some additional works on queuing with batch service are found in [20], [21], etc. One can study a queue system in two states viz.,: transient and steady state. However, most of the literature works involved steady-state only. Steady state results are inappropriate in cases where the time horizon of operation is finite. In this situation, we need time-dependent (transient state) to analyze the system behavior by tracking down the system operation at any instant of time. The transient state has been widely studied in [23], [22], and the references therein.

This paper extends the work of [18] by including the feedback policy. The inclusion of feedback, batch service and SOS makes the model more adaptable. This motivates us to explore the Marko-vian batch service queue with feedback and SOS under transient and steady state environments. The rest of the paper is organized in the following way: Section 2 contains a discussion of the model as well as the mathematical formulation of differential difference equations. We use the Laplace transform and probability generating functions with Rouche's theorem in Section 3 to develop the system transient state equations. Section 4 presents the steady-state analysis, followed by Section 5 with performance measures. Section 6 presents the numerical investigation and conclusions are made in Section 7.

II. Mathematical Formulation and Description of the Model

Consider a feedback of single batch service M/M[b]/1 queue system with SOS, in which the arrival of customers follows a Poison process with parameter A. During FES and SOS the service times are distributed exponentially with rate and , respectively. The services are rendered in batches that does not exceed the maximum capacity b. This means if the server finds the units that are equal or fewer than b in the waiting line, then he serves them all in the batch. Otherwise, if he sees in excess of b units in the waiting line, he takes a batch of b on a first-come, first-served, and the other units remain waiting in the queue. All arriving units required FES, and after a batch of units complete the FES, the batch (on the same server) may opt for SOS with the probability r0 or leave the system with probability r1. Further, if the batch is unsatisfied with the service in FES, they rejoin the queue (feedback) with probability (r2 = 1 — r0 — r1).

Model Formulation of Differential Difference Equations

The single batch service queue with SOS can be modeled by continuous time of two dimensional Markov process {(K(t), W(t)); t > 0}; where

ri hi ri hi

Figure 1: Transition State diagram for b=2

K(t) is number of units in a line at time t, W(t) is server state at time t with

W(t) — I 1, FES providing by server, ( ) \ 2, SOS providing by server.

The state-space of a Markov process is given as follows:

n — {(m, j); m > 0; j — 1,2} .

The probabilities in transient state are given by

Pm,j(t) — Pr {K(t) — m, W(t) — j}; m > 0, j — 1,2

where the probability in transient state is Pm,j(t) when m units in the waiting line at time t and

the server is rendering FES (j — 1) or SOS (j — 2).

O(t) is the probability in transient state when the waiting line is empty at time t and the server is idle. The differential difference equations for our model using Markov theory are as follows:

O' (t) = -AO(t) + rim Po,i (t) + H2 Po,2 (t), (1)

b b

P0,1 (t) = -(A + roHi + ri Hi )Po,i(t) + AO(t) + ri Hi £ Pi,i (t) + H2 E Pi2(t), (2)

i=1 i=i

PL ,i (t) = -(A + roHi + ri Hi )Pm,1 (t)+ APm-1,1 (t)+ riHi Pm+b,i(t) + H2 Pm+b,2 (t),

m > 1, (3)

P(J,2 (t) = -(A + H2 )Po,2(t)+ ro Hi Po,1 (t), (4)

Pm ,2 (t) = -(A + H2 )Pm,2 (t)+ APm-1,2(t)+ m1 Pm,1 (t), m > 1. (5)

III. The Model's Transient Solution

In this part, the Laplace transform (L.T) and probability generating function are employed to get the transient probability of the number of units in the waiting line during server idle and busy. We consider that time has been calculated from the moment that the server has taken a batch for service, without leaving none in the queue. i.e, Pq,x(0) — 1. Let us denote the Laplace transform

for O(t),Pmi(t) Pm2(t) as O*(s),P^i(s) and Pm2(s), respectively. Taking L.T of equations (1), (2), (3), (4),' (5), we get , ,

(s + a)O * (s) = ri m P*,i (s) + m Po*,2 (s), (6)

b b

(s + a + rom + ri^i)Po*,i(s) = 1 + aO *(s)+ rim E Pf*i(s) + m2 E P&(s), (7)

i=1 i=i

(s + a + ro^i + rim)Pm,i(s) = APm-i,i(s)+ rimPm+b,i(s) + mPm+b,2(s), m > 1, (8)

(s + a + ^)Po*,2 (s) = ro m Po*,i (s), (9)

(s + a + P2)Pm,2(s) = aPm-i,2(s) + romiPm,i(s), m > i. (io)

The probability generating functions are defined as:

TO TO

Pi(s,z) = E Pm,i(s)zm, P2(s,z) = E Pm,2(s)zm.

m=o m=o

Multiplying equations (7) and (8) by zm then summing over m = o to m = to, adding to (6) and re-arranging the terms, we have

zb(sO *(s) - i) + (i - zb) (rim Em=o Pm,i(s)zm + m Em-Jo Pm,2(s)zm)

P (s z) = _- ^P2 (s,z)

iv '' azb+i - (s + a + ^)zb + rm

Similarly, from equations (9) and (io), we get

(ii)

P2(s,z) = -r°miPi((s,z) ,. (i2)

2y ' Az - (s + a + m2) v '

Substituting equation (i2) in equation (ii), we have

zb(sO *(s) - i) + (i - zb) [rim Em-=io Pm,i(s)zm + v2 Em-=io Pm,2(s)z

P (s z) = _(az - (s + A + m2))

i (, ) a2 zb+2 - a(2s + 2a + ro mi + rm + v2)zb+i + (s + a ++romi + ri m )(s + a + ^2)zb

+ arimiz - (s + a + ^2)rimi - romim2

(13)

The expression for Pi (s, z) has the characteristic of converging within the unit circle. We can see that the denominator of Pi(s, z) has exactly b + 2 zeros. Applying the Rouche's theorem for denominator of Pi(s, z) to find the number of zeros on and within the unit circle of the analytic function, we observe that b of these roots lie inside or on the unit circle, one of zeros is z = 1 and others b - i, lies inside the circle and must agree with zeros of the numerator of Pi(s, z) to be converged. As a result, one zero in the denominator of Pi (s, z) is canceled by Pi(s, z) of the numerator, so that the two remaining zeros of the denominator are found outside the unit circle and let us consider them as zo and z1. Since two polynomial differ by at most a multiplicative function (constant), let us call it A(s) and is independent of z. Therefore, equation (i3) can be written as

Pi(s,z) = (az - (s + a + m2))((i - zb))A(s). (i4)

iV 7 (z - i)(z - zo)(z - zi) v '

Applying the rule of L'Hospital and setting z = 1 in (14), we obtain

P (s 1) (s + v2)bA(s) (15)

Pi(s,i)=(i - zo)(i - zi), (i5)

Taking z = i in (i2), we get

P2 (s,i) = rf Pi(s'i). (i6)

2y ' (s + m)

Applying the normalization condition

1

Pt(s,1)+ P2 (s,1) + O* (s) — ^,

we get

p (s-D=(1 zzÏTI V)2) <i7)

S(S + r0 + U2 )

Using (15) and (17)- we can determined the function A(s) as

A(s) = (1 - sO(S))(1 - Zo)(1 - z1). (18)

s(s + ro m + U2 )b

Substituting (18) into (14), we have

P {sz)— (1 - sO*(s))(1 - zo)(1 - Z1 )(Az - (s + A + m))(1 - zb) (19)

1 (,) s(s + roVt + U2)b(z - 1)(z - Zo)(z - Z1) .

When z = 0, equation (19) becomes

P* () — (1 - sO*(s))(qo - 1)(q1 - 1)(s + a + n) (20)

0,1 () s(s + rom + V2)b , ( )

where q0 — q1 — Note that P0*1(s) is the L.T of the probability of empty queue and server providing FES.

From equation (12), when z = 0 and using (20), we have

P0*2(s) — r^1(1 - sO*(s))(q° - 1)(q1 - 1) (21)

0,2 w s(s + r0m + V2)b

P*2(s) is the L.T of the probability of empty queue and server providing SOS. Using (6), (20) and equation (21), we can determine O* (s) as below:

O*(s) —_r1 m(q0 - 1)(q1 - 1)(s + A + m)r0mm(q0 - 1)(q1 - 1)_ (22)

( ) s [(s + A)(s + r0+ ^2)b + (r1 (s + A + y2)+ r0V1m)(q0 - 1)(q1 - 1)].

Equation (22) indicates the Laplace transform of the state probability that the server is idle and

the queue is empty.

IV. Steady State Analysis

Using the Tauberian property, we get the closed form expressions of the stationary probability for the number of items in the waiting line while the server is idle or active in FES and SOS. We define the stationary probabilities.

O — lim O(t) — lim sO* (s), (23)

Pm,1 — lim Pm/1(t) — lim sPm,1(s), (24)

Pm,2 — lim Pm,2(t) — limsP;„2(s). (25)

Assuming that the steady state probabilities exist, the equations (20), (21) and (22) are, respectively given by

P — (1 - O)(q0 - 1)(q1 - 1)(A + ) 0,1 (r0^1 + m)b '

r0m(1 - O)(q0 - 1)(q1 - 1)

P0-2

(rom + U2)b

O = ri yi(qo - 1)(q1 - 1)(a + u2 ) + ro U1 U2 (qo - 1)(qt - 1) A(roU1 + U2)b + (r1 u1 (A + U2) + roU1U2)(qo - 1)(q1 - 1) '

As a special case- if r2 = o, the model reduces to M/ Mb /1 queueing system with second optional service [18].

P. Vijaya Laxmi, Hasan A.B.D. Qrewi and Andwilile A. George

ANALYSIS OF MARKOVIAN BATCH SERVICE QUEUE RT&A, No 2 (68) WITH FEEDBACK AND SECOND OPTIONAL SERVICE_Volume 17, June 2022

V. Performance Measures

The measures of performance are key aspects of queueing models as they demonstrate the models efficient and effective index. In this section, we present the probability when the server is active with FES or SOS and the anticipated number of units in the waiting line during the server busy with FES or SOS.

I. The Measures of Performance in Transient State

The probability while the server is active with FES

p[FES](S)=e P^ (s)=(i s-ssO ;(;))(+ 2).

m=o s(s + ro mi + V2 )

m=o

Similarly, the probability while the server is active with SOS

P[SOS](s) = EE Pm2(s) =T(i - sO *(s)).

m=o m, s(s + ro mi + V2)

(26)

(27)

m=o

When the server is active with FES or SOS, the probability is

TO TO

Pb (s)= e pm,i (s)+ e pm,2 (s). (28)

m=o m=o

When the server is active with FES the anticipated number of items in the waiting line is

TO

E[FES](s)= E mPm,i(s).

m=o

Taking derivative of (i9) with respect to z, putting z = i, then applying the rule of L'Hospital, the expected number of units in the waiting line when the server is active with FES is obtained as

[[((s + m2)b(b - i) - 2ab](qo - i)(qi - i) - [(s + m2)b(4qoqi - 2(qo + qi))]] ETO p* ( ) =_(i - sO *(s))

m=o m m,i(s)= 2b (s(s + romi + m2)(qo - i)(qi - i)) .

(29)

Similarly, from equations (22) and (29), the anticipated number of units in the waiting line when the server is active with SOS is given by

TO

E[SOS](s) = E mPm,2(s)

m=o

[[(b(b - i) - 2ab](qo - i)(qi - i) - [b(4qoqi - 2(qo + qi))]]

=_ro mi (i - sO * (s))

2b (s(s + ro mi + m2 )(qo - i)(qi -1))

+ Aro mi(i - sO *(s)) . (3o)

s(s + m2)(s + romi + m) The overall queue length is given by

TOTO

Lq(s) = E mPm,i(s) + E mPm,2(s) = E[FES](s) + E[SOS](s). (31)

m=i m=i

The overall amoMnt of time a Mnit spends in the waiting line is determined by

W (s) = ETO=1 mPm,i(s) + EETO=1 mPm,2 (s) = E[FES](s) + £[sos](s). (32)

II. The Measures of Performance in Steady State

Suppose the limit of (23), (24) and (25) exist, the steady state quantities from (26) to (32), respectively are denoted by

P[FES] P[SOS] Pb

E mPm,1

m=0

TO

E mPm,2

m=0

Wq

E P1 = a - o)p2 n=0 ro ^ + E P = rom(i - O)

E 1 m,2 — . ,

m=o ro m+m

TO TO

E, Pm,1 + Pm,2,

m=o m=o

[[^b(b - 1) - 2Ab](ro - 1)(ri - 1) - [^b(4rori - 2(ro + ri))]]

_(1 - O)

2b (rom + m)(ro - 1)(r1 - 1)) '

[[b(b - 1) - 2Ab](qo - 1)(q1 - 1) - [b(4qo?1 - 2(qo + q1))]] ro^(1 - O) 2b fa (rom + rn)(qo - 1)(q1 -1))

aro^1(1 - o) m (rom + m)'

TOTO

E mPm,1 + E mPm,2 = E[FES] + E[SOS],

m=o m=o

ETO=1 mPm,1 + ETO=1 mPm,2 _ E[FES] + E[SOS]

+

a

a

(33)

VI. Numerical Results and Discussion

In this part, the numerical examples are presented after inverting L.T of equations (2o), (21), (22) and (26) to (32) into time domain with the help of Mathematica software. Since an inverted L.T expressions are too long, therefore, we compute the model numerically by using the arbitrary model parameters as A = 2, = 4.5, = 3.5, ro = o.36, b = 5, r1 = o.4. The effect of these parameters on probability and other performance measure with respect to time are shown in terms of graphs and tables.

Table 1: The effect ofro on r\, LqFES, LqSOS and Lq with A = 2, = 4.5, ^ = 3.5, r2 = o.1, b = 5.

ro r1 LqFES LqSOS Lq

o.1 o.8 1.61413 o.136298 1.75o42

o.2 o.7 1.529o5 o.2583o6 1.78736

o.3 o.6 1.44568 o.366441 1.81213

o.4 o.5 1.36723 o.462234 1.82947

o.5 o.4 1.29436 o.54716 1.84152

o.6 o.3 1.22737 o.622792 1.85o16

o.7 o.2 1.166o2 o.69o48 1.8565

o.8 o.1 1.1o997 o.751454 1.86143

TO

L

q

Table 1 depicts the effect of the probability of opting for SOS (ro) on the expected queue length (Lq). We observe that for fixed feedback probability (r2), an increase in ro leads to a decrease in the probability of the departure (r1), resulting in decrease of LqFES while LqSOS and overall Lq increases. This is because as ro increases, more customers attend for SOS, resulting in increasing the Lq in SOS and overall queue size, as we expect.

Table 2: Effect of/eedback probability fa) on ro, LqFES, LqSOS and Lq with A = 2, y1 = 4.5, y2 = 3.5, ri = 0.2, b = 5

r2 r0 LqFES LqSOS

L

q

0.2 0.3 0.4 0.5 0.6 0.7

0.6 0.5 0.4 0.3 0.2 0.1

1.29082 1.44771 1.65249 1.93281 2.34909 3.05593

0.663747 0.630166 0.586778 0.527648 0.441877 0.30109

1.95457 2.07788 2.23927 2.46045 2.79096 3.35702

Table 2 shows the effect of feedback probability (r2) on the expected queue length. For the fixed probability of the departure (ri), as r2 increases, the probability of opting for SOS (r0) decreases, which leads to decrease LqSOS and opposite trend observed in LqFES and overall Lq. This implies that as r2 increases more customers are not satisfied with service (FES). Therefore, the customers feedback by joining the queue tends to increase LqFES and overall queue size.

Table 3: The effect of r1 on feedback probability r2, LqFES, LqSOS and Lq with A = 2, ¡1 = 4.5, ¡2 = 3.5, r0 = 0.1, b = 5

r1 r2 LqFES LqSOS Lq

0.1 0.8 1.61413 0.136298 1.75042

0.2 0.7 1.52905 0.258306 1.78736

0.3 0.6 1.44568 0.366441 1.81213

0.4 0.5 1.36723 0.462234 1.82947

0.5 0.4 1.29436 0.54716 1.84152

0.6 0.3 1.22737 0.622792 1.85016

0.7 0.2 1.16602 0.69048 1.8565

0.8 0.1 1.10997 0.751454 1.86143

From Table 3, for the fixed probability of opting for SOS r0, as r1 increase, r2 decrease, resulting in decreasing LqFES, LqSOS and overall Lq, which agrees with our intuition.

Table 4: The effect of ¡1 on the expected queue length with different values of y 2 and r0 = 0.36, r1 = 0.4, A = 3.8.

1 y2 = 1.8 y 2 = 2 y2 = 2.2

2.4 6.54206 6.09142 5.7638

2.8 5.71485 5.31505 5.02568

3.2 5.23358 4.85613 4.57894

3.6 4.92135 4.55096 4.27954

4.0 4.7030 8 4.33623 4.06484

4.4 4.54338 4.17542 3.90165

The impact of y 1 on the expected queue length (Lq) with different values of y2 is shown in Table 4. Here Lq deceases as 1 increases, as we expect. Furthermore, for fixed 1, as 2 increases, Lq decreases. The reason is that, by increasing y1( y2), customers are serviced faster and results in reducing the length of the queue.

Figure 2: The effect of departure probability r1 on the Lq

Figure 2 shows the effect of departure probability r1 after completing FES on the expected queue ( Lq). We observe that Lq decreases with time until it reaches a steady state. Further, we notice that as r1 increases , Lq decreases. The reason is that more customers depart from the system after completion of FES, hence reducing Lq.

Figure 3: The effect of A on the expected Lq «s time progresses

Figure 3 display the effect of the rate of arrival (a ) on the expected queue length (Lq). As time progresses, Lq decreases until it attains steady state. Moreover, as a increases, more customers enter the queue, resulting in increase in Lq.

Figure 4: The effect of arrival rate on Lq with different batch size

Figure 4 displays the effect of a on the expected length of queue (Lq) with different batch size. For fixed a, we observe that as b increases, Lq decreases. This is because, more customers are served in batch at a time, which results in reducing the queue length. Further, as b keeps constant, Lq increases as a increases, as intuitively expected.

2.5

§ 2

0) 3 cr

aa 1.5

a E

a>

c 0.5

0 0.1

1 1 1

/

..a.W FES q / /

/ /

SOS / J

<1

- -..Overall W q

„**........♦

_______

0.2 0.3 0.4 0.5 0.6 Probability of feedback (r )

0.7 0.8

Figure 5: The effect of r2 on Wq

The effect of the probability of feedback (r2) on the expected waiting time in the queue (Wq) is presented in Figure 5 . We can observe from the graph that as r2 increases, WqFES, WqSOS, and overall Wq increase. Because as r2 increases, more customers rejoin the queue (feedback), resulting in increasing WqFES, WqSOS and overall Wq.

VII. Conclusion

We investigate the transient and steady state behavior of a single server batch service queue with SOS and feedback. We use the probability generating function, Laplace transforms, and Rouche's theorem to obtain the transient state probabilities after inverting Laplace transform into the time domain. Also, the Tauberian theorem is applied in Laplace transform expression to obtain the steady state probabilities. Finally, we present numerical results as tables and figures to show the effects of various parameters on the model performance measures.

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