Heavy outgoing call asymptotics for retrial queue with two way communication and multiple types of outgoing calls Текст научной статьи по специальности «Математика»

Discrete & Continuous Models 2019 27 (1) 5—20

. „ . http://journals.rudn.ru/miph

& Applied Computational Science

Queuing theory

Research article

UDC 519.872

DOI: 10.22363/2658-4670-2019-27-1-5-20

Heavy outgoing call asymptotics for retrial queue with two way communication and multiple types

of outgoing calls

Anatoly Nazarov, Svetlana Paul, Olga Lizyura

Institute of Applied Mathematics and Computer Science National Research Tomsk State University 36 Lenina ave., Tomsk, 634050, Russian Federation

(received: December 7, 2018; accepted: October 21, 2019)

In this paper, we consider a single server queueing model M|M|1|N with two types of calls: incoming calls and outgoing calls, where incoming calls arrive at the server according to a Poisson process. Upon arrival, an incoming call immediately occupies the server if it is idle or joins an orbit if the server is busy. From the orbit, an incoming call retries to occupy the server and behaves the same as a fresh incoming call. The server makes an outgoing calls after an exponentially distributed idle time. It can be interpreted as that outgoing calls arrive at the server according to a Poisson process. There are N types of outgoing calls whose durations follow N distinct exponential distributions. Our contribution is to derive the asymptotics of the number of incoming calls in retrial queue under the conditions of high rates of making outgoing calls and low rates of service time of each type of outgoing calls. Based on the obtained asymptotics, we have built the approximations of the probability distribution of the number of incoming calls in the system.

Key words and phrases: retrial queueing system, incoming calls, outgoing calls, asymptotic analysis method, Gaussian approximation

1. Introduction

Retrial queueing systems are characterized by the following distinctive feature: a customer who cannot receive service remains in the system and tries to occupy the server after some random delay. The pool of unsatisfied customers is called the orbit. Retrial queues have applications in telecommunication, computer networks and in daily life [1,2].

In retrial queues idle time of the server is the downtime and it should be reduced to increase the efficiency of the system. We consider systems where operator not only receives calls from outside but also makes outgoing calls

© Nazarov A., PaulS., LizyuraO., 2019

This work is licensed under a Creative Commons Attribution 4.0 International License http://creativecommons.org/licenses/by/4.0/

in the idle time. In queueing theory a model with this feature have been considered previously [3]. However, the retrial behaviour of customers is not taken into account. In call centers operators could receive arriving calls but as soon as they have free time and are standby mode they could make outgoing calls [4-7]. Systems with this server behaviour are called retrial queues with two way communication. Retrial Queues with two way communication have been studied recently [8-11]. In these papers Markovian models with two way communication were considered. Model of retrial queue with two way communication and multiple types of outgoing calls was considered by Sakurai and Phung-Duc [12]. For this model numerical algorithm of calculating joint stationary distribution of system state was obtained. Multiserver retrial queue with two way communication was studied in [13]. Recently the two way communication retrial queues with finite source [14], with server-orbit interaction [15,16], with finite orbit [17], with breakdowns [18] and with a constant retrial rate [19] were considered.

Asymptotic analysis methods have applications in queueing theory. Nazarov, Paul and Gudkova propose an asymptotic analysis method to research M | M |1|1 retrial queue with two way communication under low rate of retrials condition [20]. Nazarov, Paul and Phung-Duc extended this model to MMPP|M1111 retrial queues and derived asymptotics in heavy outgoing call conditions [21].

In this paper, we consider retrial queue with two way communication and multiple types of outgoing calls. We assume that each type of outgoing calls has different rate and service times follow distinct exponential distributions. The main aim of this paper is to derive asymptotics for the model in two limit conditions: i) high rate of outgoing calls and ii) low service rate of outgoing calls. In both cases, the number of incoming calls in the system increases.

The rest of the current paper is organized as follows. In Section 2 and 3, we describe the model in detail and preliminaries for later asymptotic analysis. In Section 4 and 5, we present our main contribution to the model. In Section 6 we show the ranges of parameters under which our approximations are usable. Section 7 is devoted to concluding remarks.

2. Model and preliminaries 2.1. Model description

Figure 1 shows the structure of the model.

We consider a single server retrial queue with two way communication and multiple types of outgoing calls. Incoming calls arrive at the system according to a Poisson process with rate A and try to occupy the server for an exponentially distributed time with rate ¡i1. Incoming calls that find the server busy join the orbit and repeat their request for service after an exponentially distributed time with rate a. When the server is idle it makes an outgoing call of type n in an exponentially distributed time with rate an. There are N types of outgoing calls whose durations follow N distinct exponential distributions. We assume that the durations of outgoing calls of type n follow the exponential distribution with rate

A ßi

ßn

ar,

Figure 1. Markovian retrial queue with two way communication and multiple types

of outgoing calls

2.2. Problem definition

Let k(t) denote the state of the server at the time t ^ 0,

0, if the server is idle, k(t) = i 1, if an incoming call is in service,

n, if an outgoing call of type n is in service, n = 2, N + 1.

Let i(t) denote the number of incoming calls in the system at the time t. It is easy to see that process {k(t), i(t)} forms a continuous time Markov chain. We assume that the Markov chain is ergodic and the stationary distribution of {k(t), i(t)} exists.

Let P{k(t) = k, i(t) = i} = Pk(i) denote the stationary probability distribution of the system state which is the unique solution of Kolmogorov system of equations:

N+1

A + ia + ^ a„,

n=2

N +1

Po(i) + ßiPi(i + 1) + J] ßnPn(i) = 0,

n=2 (1)

- (A + ßi)Pi(i) + APi(i - 1) + APo(i - 1) + iaPo(i) = 0,

{- (A + ßn)Pn(i) + APn(i - 1) + anPo(i) = 0, n = 2, N + 1.

Let Hk(u) denote the partial characteristic functions Hk(u) = ejuiPk(i), _ i=0

k = 0, N +1, where j = yf—T. Multiplying equations of system (1) by ejui

and taking the sum over i yields

N+i

A+

an

n=2

N +i

Ho(u) + jaH0(n) + ßie-juHi(n) + ^ ßnHn(u) = 0

n=2

— (A + ßi)Hi(n) + AejuHi(n) + AejuHo(n) - jaH'0(n) = 0,

[ - (A + ßn)Hn(u) + AejUHn(u) + anHo(u) = 0, n = 2, N + 1.

The characteristic function H(u) of the number of incoming calls in the retrial queue is expressed through partial characteristic functions Hk (u) by

N +1

H(u) = Hk (u). The main content of this paper is the solution of system

k=0

(2) by using an asymptotic analysis methods in two limit conditions: of the high rate of making outgoing calls and the low rate of service time of outgoing calls.

3. Prelimit analysis

In this section, we obtain expressions for the stationary distribution using the characteristic functions. First, we derive explicit expression for the characteristic function H(u) of the number of incoming calls in the system.

Theorem 1. Explicit expression for the characteristic function H(u) of the number of incoming calls in M|M|1|N retrial queue is given as follows:

H(u)

1 + Vi

N +1

1 + E

a

n=2

7n + A(1 - eJU)

x

1 - P

1 - peJU

x

(1+v2)+1 N+1

n

n=2

1 - Pn

1 - Pneju

a A)

a6ri

where

Pn

P =

A 71

A

7n + A

N +1

N+1

ak ak

V1 = 2_. 77' V2 = ^ —,

k=2

7k

k=2

9k

On = A + 7n - 71, n = 2,N + 1.

Proof. From equations 2 and 3 of the system (2) we obtain expressions for partial characteristic functions:

H1(u)

AeJU

71 + A(1 - eJU)

^ Ho(u) -

Ja

71 + A(1 - eJU)

H0 (u),

(3)

Hn(u)

an

7n + A(1 - eJU)

H0(u), n = 2,N + 1.

(4)

Substituting this equations into the first equation of the system (2), we find that

H0 (u) = ja

AeJU

71 - AeJU

+

71 + A(1 - eJU) N+1

ane

JU

71 - AeJU

n=2 7n + A(1 - eJu)

H0(u). (5)

1

The solution of this differential equation is given by

Ho(u) = ro

1 - P 1 - peju

C(1+V2) N+1

n

n=2

1 - Pn 1 - Pneju

(6)

A

Ho(0) = P{k(t) = 0}, V2

where p = —, r0

71 _

On = A + 7n - 71, n = 2, N + 1.

Substituting u = 0 into the system (2) yields:

N +1

N+1 a fc=2

A

7n + A

N +1

A + ^ an r0 + jV H0(u)|u=0 + X] 7krk = 0'

n=2

fc=1

- 71r1 + Ar0 - jV H0(u)|u=0 = 0

(7)

7nrn + anr0 = 0, n = 2, N + 1,

where expression for H0(u)|U=0 can be obtained substituting u = 0 into (5). It follows from equations 2 and 3 of the system (7) that

r1

N+1

A

■ , — 71 71 - A V 71 n=2 7n

AA

— +

r0,

a

rn = — r0, n = 2, N +1.

7n

Furthermore, from the normalization condition: rk = 1, we obtain

A

r0

N +1 fc=0 an(71 - A)

71 - A _ a _ an

71 (1 + V1)' 71' n 717n (1 + V1)

n = 2,N + 1,

N+1

ak

where v1 = ^ —. Substituting (6) into (3) and (4) and summing up results,

k=2 7k

we obtain

H (u)

1

1 + V1

N+1

1 + £

an

n=2

7n + A(1 - eju)

1 - P

1 - peju

x

(1+V2) + 1 N+1

n

n=2

1 - Pn

1 - Pneju

an (An- A) aOn

a

n v^n

4. Asymptotic analysis of the model under the high rate of making outgoing calls

In this section, we will investigate system (2) by asymptotic analysis method under the high rate of making outgoing calls condition. In particular, we prove that asymptotic characteristic function of the number of incoming calls in the system corresponds to Gaussian distribution.

Denoting an = aYn, we obtain

N +1

A + a

n=2

N+1

Ho(u)+ jaHO(u) + ^e-jUHi(u) + J] ^Hn(u) = °,

n=2

- (A + ^i)Hi(u) + AejuHi(u) + AejuHo(u) - jaH(u) = 0 k - (A + ^n)Hn(u) + AejuHn(u) + aYnHn(u) = (

(8)

n

2, N + 1.

4.1. First order asymptotic

Theorem 2. Suppose i(t) is the number of incoming calls in the system of the stationary M|M|1|N retrial queue with outgoing calls, then the (9) holds

lim Eejw^ = ejwK1, (9)

where

, N+i

M^i ST^Yn ,1fV.

- A) n=2 ^n

Proof. We denote a = 1/e in the system (8), and introduce the following notations

u = ew, H0(u) = eF0(w,£), Hk(u) = Fk(w,e), k =1,N + 1, in order to get the following system

N+i dF (w e) - (Ae + J] Yn)Fo(w,e) + ja °( , ) + ^e-jW£Fi(w,e)+

Yn)Fo(w,e)+ ja-^^

n=2

N+i

+ ^nFn(w,e) = 0,

n=2

- (A + ^i)Fi(w,e) + Aejw£Fi(w,e) + Aejw£eFo(w,e)-

- (A + ^n)Fn(w, e) + Aejw£Fn(w, e)+

_ .^dFo(w,e) = ° j dw ,

+ YnFo(w,e) = °, n = 2, N + 1.

Summing up equations of system (11), we obtain

N+1

AeF0(w,£) + (A - 71e-JW£)F1(w,e) + A > Fn(w,e) = 0

n

n=2

(12)

Considering the limit as e ^ 0 in the system (11) and equation (12), then we will get

N+1

N+1

X YnF0(w) + jaF0 (w) + X 7kFk(w) = 0

n=2

k=1

- 71F1(w) - jaF0(w) = 0,

- 7nFn(w)+ YnF0(w) = 0, n = 2,N + 1,

N +1

- (71 - A)F1(w) + A E Fn(w) = 0.

n=2

(13)

We propose to get the solution of the system (13) in the form of

Fk(w) = $(w)rk, k = 0, N +1.

(14)

Here rk, k = 1,N + 1 is the probability of the server state k; r0 has no sense of probability, since the probability that the server will be in the zero state as a is zero:

N+1

X Ynr0 + Ja^7-yr0 + ^ 7krk =

N+1

n=2

$(w)

k=1

• ^'(w) n

71r1 - = 0

- 7nrn + Ynr0 = 0, n = 2, N + 1,

N+1

- (71 - A)r1 + A E rn = 0.

n=2

(15)

$'(w)

As the relation j——- does not depend on w, the function is obtained in

$(w)

the following form $(w) = exp{jwK1}, which coincides with (9). The value of the parameter k1 will be defined below. We rewrite the system (15) in the form

N+1 £

N +1

Ynro - Kifoa

+ XI 7fcrfc = 0,

- 7iri + Kiroa = 0,

(16)

- + Ynro = 0, n = 2, N + 1,

N+1

- (71 - A)ri + A X rn = 0.

n=2

The normalization condition for stationary server state probability distri-

N +1

bution is rk = 1. We have

fc=i

' - 7nrn + Ynro = 0, n = 2, N + 1,

N +1

- (71 - A)ri + A X rn = 0,

n=2

(17)

N+1

£ rk = 1.

fc=i

The solution of the system (17) is given by

7i - A A Yn(7i - A)

ro = -, ri = -, rn = -.

7iv 71 7n7ivi

n = 2,N + 1,

(18)

N +1

Y

where v1 = ^ —. Substituting (18) into system (16), we obtain an equation

n=2 7n

for k1, which coincides with (10).

The first order asymptotic i.e. Theorem 2, only defines the mean asymptotic value K1a of a number of incoming calls in the system in prelimit situation of a —^ to. For more detailed research of number i(t) of incoming calls in the system let's consider the second order asymptotic. □

4.2. Second order asymptotic Theorem 3. In the context of Theorem 2 the following equation is true

. — - (jw)2

lim E exp < jw a _— } = e 2 I J a

K2

(19)

where

A 71(71 - A)(AV2 + vi) + A2vi

K2 =---/-TT2-, V1

a (71 - A)2

N+1

Yn _, V2 7n

N+1

£7!. (20)

n=2

n=2

Proof. We introduce the following notations in the system (8)

Hk(—) = exp{juaKi}Hk2)(u), k = 0, N + 1, (21)

and we get

N+1 \ Tt(2) / \ • dHn )(

; Yn + aaKJ H0 y (—) + ja——

n=2 /

N +1

+ ^ie-jUH;2)(—) + £ frnHn2)(—) = 0,

n=2

(A + fri)H(2)(—) + AejUH(2)(—) + (Aeju + aaKi)H,12)(—)- (22)

r(2)

- jadHO—)=0,

d—

- (A + frn)Hn2)(—) + AejUHn2)(—)+ t(2)I

+ aYnH02)(—) = 0, n = 2, N + 1.

Denoting a = 1/e2, and introducing the following notations

— = we, H,S2)(—) = e2F0(2)(w,e), Hf (—) = F(2) (w, e), k = 1, N + 1,

we obtain

Jae

dF0(2)(w,e)

dw

N+i

aKi + Ae2 + ^ Yn ) F(2)(w, e)+

n=2 N +i

+ frie-j'weF(2) (w, e) +£ frnF12) (w, e) = 0

n=2

- (A + fri)F(2)(w,e) + AejW£F1(2)(w,e)+

(2),... ^ ,_dFo(2)(w,e)

(2)/

+ (AejW£e2 + aKi)F0(2)(w, e) - jae

- (A + frn)Fl2)(w,e) + AejW£Fl2)(w,e)+ ,(2)

dw

+ YnF0(2)(w,e) = 0, n = 2, N + 1.

(23)

(24)

Summing up equations of the system (24), we obtain

N+i

Ae2F0(2)(w,e) + (A - frie-jW£)Fi(2)(w, e) + A V F(2)(w,e) = 0.

n=2

(25)

0

Our idea is to seek for a solution of the system (24) and equation (25) in the form

P(2)

F(2)(w,e) = $2(w){rfc + jwe/} + o(e2), k = 0,N + 1. (26)

Substituting (26) to (24) and (25), laying out the exhibitors in tailor series and taking (16) into account, dividing these equations by e and taking the limit as e — 0, we have

N+1 \ N+1 , , ,

- I frKi + Yn I f0 + y, 7 fc/fc - 7iri + fr \ r0 = 0,

) t! w$(w) (w)

fr«i/o - 7i/i + Ari - fr 2 ro = 0,

w$(w)

-7n/n + Arra + Yn/o = 0, n = 2, N + 1,

N+1

-(71 - A)/i + A X /n + 7iri = 0.

y n

n=2

(w)

This equations imply that ——r doesn't depend on w and thus the

w$2(w)

function $2(w) is given in the following form

„ ^ i (jw)2

$2(w) = exp j -K2

which coincides with (19). We have

$2(w)

w$2(w)

and then we obtain the system

-K2

N+1 \ N+1

- I frKi + X Yn I /o + X 7 fc/fc = 7iri + frK2ro,

n=2 / fc=l

frKi/o - 7i/i = -Ari - frK2ro,

- 7n/n + Yn/o = -Arn, n = 2, N + 1,

N+i

- (7i - A)/i + A X ^ = -7iri.

(27)

n=2

Substituting values (18) into the system (27), we have

Yn , , A(7i - A)Y

/n = -/ + V ^ 2 7 'n, n = 2, N +1,

7n 7i7nvi

, _ Avi A2Vo A /1 —-T /0 +---r

" — A ^1v1 " — A where

N+1 N+1

Yk Yk

v1 — ~, v2 — "2 . k=2 "k k=2 "k

Substituting this expressions into equation 2 of the system (27), we obtain an equation for k1, which coincides with (20).

Second order asymptotic i.e. Theorem 3, shows that the asymptotic probability distribution of the number i(t) of incoming calls in the system is Gaussian with mean asymptotic K1a and variance k2a. □

5. Asymptotic analysis of the model under the low rate of service time of outgoing calls

In this section, we will investigate system (2) by asymptotic analysis method under the low rate of service time of outgoing calls condition.

Denoting "n — "Yn, we obtain

N+1

an

n=2

a+E

N+1

Ho(u)+jVtf£(u) + "1e-jUH1(u)+^ J] YnHn(u) — 0,

n=2

— (A + "1)H1(u) + Aeju H1(u) + AejuHo(u) — jaHO (u) — 0,

(28)

{— (A + "Yn)Hn(u) + AejUHn(u) + anHn(u) — 0, n — 2, N +1.

Theorem 4. Suppose i(t) is a number of incoming calls in a system of stationary M|M|1|N retrial queue with two way communication, then the following equation is true

i N+1 a N+1 / A \ - ^ny

H(u) — lino Ee^ — 1 £ -j 1 — jw^ , (29)

n=2 Yn — j-A n=2 V Y

N+1 a

n

where v1 — —.

n=2 Yn

Proof. We denote " — e, let's substitute the following in the system (28)

u — we, H0(u) — eF0(w,e), Hk(u) — Fk(w,e), k — 1,N + 1.

We will get the system

/ N+1 \

- (a + X anj

. „ , , . dFo(w,e) a« eFo(w,e) + ja----+

dw

N +1

+ 7ie-jW£Fi(w, e) + e J] 7„F„(w, e) = 0,

n=2

- (A + 7i)Fi(w,e) - jadFo(w,e) + AejW£Fi(w,e) +

dw

+ AeejW£Fo(w,e) = 0,

- (A + e7ra)F„(w, e) + AejW£F„(w, e)+

+ a«eFo(w, e) = 0, n = 2,N + 1.

(30)

Considering the limit as e — 0 in the system (30) then we will get

-jfrFq(w) - 7iFi(w) = 0, jVF^(w) + 71F1 (w) = 0.

(31)

Summing up equations of the system (30) we have

N+i

AeFo(w, e) + (A - 7ie-jW£)Fi(w, e) + A ^ 7„F„(w, e) = 0. (32)

n=2

Laying out the exhibitors in tailor series, dividing equations by e and taking the limit as e — 0, taking (31) into account, we obtain

N +i

N +i

A + X a« Fo(w) - jw^iFi(w) + ^ Y«Fra(w) = 0,

n=2

n=2

-jaF^(w) - 7iFi(w) = 0,

(Ajw - 7n)Fra(w) + a«Fo(w) = 0, n = 2, N + 1

N +i

-(7i - A)Fi(w) + A J] F«(w) = 0

n=2

From the last system of equations we have

F« (w) =

Fi(w) =

a«

7„ - jwA

Fo (w),

A

N+i

- A

X F«(w)

n=2

(33)

(34)

Then

x N+1

Fi(w) =-AFo(w) £-=—. (35)

- A Yn - J-A

n=2

Substituting (35) into (31), we obtain

N +1

A N +1

A^1 ^ v^ a

F(-)= J, ,, Fo(-)

- A) n=2 Yn- j-A

The solution of differential equation is given by

N +1 x , x--

n=2

Yn

Fo(-) = 1 - j-- , (36)

where C is an integration constant and its value will be obtained later. We

N +1

denote asymptotic characteristic function Fk (w) — $(w). Substituting

k=1

(36) into (33) and (34), we obtain

\ N+1 N+1 , , \--^

F 1 (w) — -- > -:—tC 1 — jw—

"1— A Yk— JwA k=2 V Yk

a + / A \ ^ (pI - _

F"<w' — Y-jAC lit1 — J"wYj • n — 2'N + 1-

Summing up equations, we have

N+1 / A \--N+1

t / \ "1 ^TTii • M CT(p1-A) V^ an

$(w) — ^rr CH 1 — jw—

^- A n=2 V W n=2 Yn - j-A

Using condition $(0) = 1, we obtain

A N+1

— A , V^ an

C =-, where v1 = > —.

^ n^ Yn

We obtain the characteristic function (29). □

6. Approximation accuracy

The accuracy of the approximation P(2)(i) is defined by using Kolmogorov

range A2 = max

0<i<N

E (P(v) - P(2)(v))

v=0

which represents the difference be-

tween distributions P(i) and P(2)(i), where P(i) is obtained by using

inverse

Fourier transform for the characteristic function of the M|M111N retrial queue and the approximation P(2) (i) is given by obtained asymptotics. We consider N = 3, A = 0.2, 71 = 1 and fr =1 for Tables 1 and 2.

Table 1

Kolmogorov range, = 2, = 3, =4, 72 = 1, 73 = 2, 74 = 3

a = 3 a = 5 a = 10 a = 50 a = 100

A2 0.066 0.043 0.023 0.01 0.007

Table 2

Kolmogorov range, 72 = 2, 73 = 3, 74 = 4, a2 = 1, «3 = 2, a4 = 3

7 = 0.05 7 = 0.035 7 = 0.02 7 = 0.01

A2 0.059 0.044 0.026 0.014

7. Conclusions

In this paper, we have considered retrial queue with two way communication with multiple types of outgoing calls. We have found characteristic function of the number of incoming calls in the system. We have found the first and the second order asymptotics of the number of calls in the system under the condition of the high rate of making outgoing calls. Based on the obtained asymptotics we have built the Gaussian approximation of the probability distribution of the number of incoming calls in the system. We have found asymptotic characteristic function of the number of incoming calls in retrial queue under the condition of the low service rate of outgoing calls. In future we plan to consider this retrial queueing system under other asymptotic conditions.

Acknowledgments

This research was financially supported by Russian Foundation for Basic Research (the Agreement number 18-01-00277).

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For citation:

A. Nazarov, S. Paul, O. Lizyura, Heavy outgoing call asymptotics for retrial queue with two way communication and multiple types of outgoing calls, Discrete and Continuous Models and Applied Computational Science 27 (1) (2019) 5-20. DOI: 10.22363/2658-4670-2019-27-1-5-20.

Information about the authors:

Anatoly A. Nazarov (Russian Federation) — Professor, Doctor of Technical Sciences, Head of Department of Probability Theory and Mathematical Statistics, Institute of Applied Mathematics and Computer Science, National Research Tomsk State University (e-mail: nazarov.tsu@gmail.com, phone: +7(3822)529599, ORCID: https://orcid.org/0000-0002-2091-6011, ResearcherID: O-5862-2014, Scopus Author ID: 7201780364) Svetlana V. Paul (Russian Federation) — Candidate of Physical and Mathematical Sciences, Assistant Professor of Department of Probability Theory and Mathematical Statistics, Institute of Applied Mathematics and Computer Science, National Research Tomsk State University (e-mail: paulsv82@mail.ru, phone: +7(3822)529599, ORCID: https://orcid.org/0000-0002-3681-0676, ResearcherID: E-7860-2017, Scopus Author ID: 57031144200)

Olga D. Lizyura (Russian Federation) — Master's Degree Student of Institute of Applied Mathematics and Computer Science, National Research Tomsk State University (e-mail: oliztsu@mail.ru, phone: +7(999)4996743, ORCID: https://orcid.org/0000-0003-4463-4876, ResearcherID: G-7896-2019)