Asymptotic analysis of retrial queueing system M/M/1 with impatient customers, collisions and unreliable server Текст научной статьи по специальности «Математика»

Journal of Siberian Federal University. Mathematics & Physics 2020, 13(2), 213—217

DOI: 10.17516/1997-1397-2020-13-2-213-217 УДК 539.374

Anisotropic Antiplane Elastoplastic Problem

Sergei I. Senashov* Irina L. Savostyanova*

Reshetnev Siberian State University of Science and Technology

Krasnoyarsk, Russian Federation

Olga N. Cherepanova*

Siberian Federal University Krasnoyarsk, Russian Federation

Received 10.11.2019, received in revised form 11.01.2020, accepted 20.02.2020 Abstract. In this work we solve an anisotropic antiplane elastoplastic problem about stress state in a body weakened by a hole bounded by a piecewise-smooth contour. We give the conservation laws which allowed us to reduce calculations of stress components to a contour integral over the contour of the hole. The conservation laws allowed us to find the boundary between the elastic and plastic areas. Keywords: anisotropic elastoplastic problem, antiplane stress state, conservation laws.

Citation: S.I.Senashov, I.L.Savostyanova, O.N.Cherepanova, Anisotropic Antiplane Elastoplastic Problem, J. Sib. Fed. Univ. Math. Phys., 2020, 13(2), 213-217.

DOI: 10.17516/1997-1397-2020-13-2-213-217.

Introduction

Fields of shifts and stresses in the case under consideration are the following [1]

u= v =0, w = w (x,y) ax = ay = az = rXy = 0, rxz = t1 (x,y), TyZ = t2 (x,y) . (1)

Here u, v, w are shift vector components, ax, ay, az, Txy, txz, Tyz are stress components, x, y, z the Cartesian coordinates, axis directed parallel to the element.

In the elastic zone there are the relations

дт1 дт2 , — 1—-— = 0 (equilibrium equation), dx dy (2)

1 dw 2 dw , т = —— ,t = G2^— (Hooke s law) . dx dy (3)

Here Gi are constants called elastic moduli [2]. From (2), (3) there arise relations in the elastic zone G. T2 + G2 fw =0. dx2 dy2 (4)

G дт1 G дт2 g2— = g1^—. дy дx (5)

*sen@mail.sibsau.ru https://orcid.org/0000-0001-5542-4781 ^ruppa@inbox.ru https://orcid.org/0000-0002-9675-7109 ^cheronik@mail.ru © Siberian Federal University. All rights reserved

From (2) and (5) it follows that

.1 ^2

satisfy the system of linear equations

Fi

дт1 дт2

дх + dy

0, F2

дт1 дт2

ду дх

where n = G1/G2.

In the plastic zone there holds the relation (2), and also

(6)

ai3(т1)2 + «23(т2)2

1 (yield condition),

2 3w дх

i д'ш

ду

(Hencky's equation) .

(7)

(8)

Here a13,a23 are constants called anisotropy coefficients.

On the boundary of the elastic and plastic areas the stresses and shifts are supposed to be continuous.

1. Conservation laws

By a conservation law for the system of equations (6) we shall call the relation of the form of дЛ(х,у,т 1,т 2) дБ(х,у,т 1,т 2

дХ + ду

where шг = шг(х,у,т 1,т2) are some functions not identically zero simultaneously.

Note. A more general definition of conservation laws and their use in mechanics of a solid body being deformed can be studied for example in [3-5].

For the purposes that are set in this article a simplified formulation in the form of (9) will suit fine.

In (9) the values Л, B are called conserved current components.

Let us assume that the components Л, B appear as follows

Л = а1т1 + в1т2 + Y1, B = а2т1 + в 2т2 + Y2, (10)

= ш1 F1 + ш2 F2,

(9)

where аг = аг (х, у), вг = вг (х, у), y1 = Yг (х, у) are some smooth functions to be determined. Let us substitute (10) into (9), as a result we obtain

ахт + а тх + вхт + в тх + Yx + аут + ату + вут + в ту + Yy =

= j1 (тХ + тУ) + j2 (тУ- птХ) =

where the index below stands for a derivative with respect to the corresponding variable. From (11) we obtain

а1 = ш1,

а1 2 2 2

в = —nj , а = ш

From (12) excluding ш1 we obtain

а1 = в2, в1 = no^, ахх — ney =0, в1 + а1 = 0, yI + Y^l

By virtue of relations (12) the conserved current components are written as

—в1 1

0.

Л = а1т1 + в1т2 + Y1, B

-т1 + а1т2 + Y2 ■

(11)

01 = а1 + ау =0, fi1 + вУ = 0, yI + дУ, = °- (12)

(13)

(14)

n

Since the right-hand part (9) is equal to zero, according to Green’s formula we obtain / / (Ax + By) dxdy = (p Ady — Bdx =

JJs JdS /1Гч

f f— в1 \ (15)

= j) (a1 t 1 + в1 t2 + Y1 )dy — I-t 1 + а1 т2 + yj dx = 0,

where S is the area, dS is its piecewise-smooth boundary. All the functions in (15) are supposed to be smooth.

2. Elastoplastic problem for an arbitrary hole in case when the plastic area surrounds the entire hole

Assume C is a piecewise-smooth contour, there is a load applied to it

l1 т1 + I2T2 = Tn, \Tn | ^

l2 a23 + l2 a13 a13 a23

(16)

where (l1 ,l2) are normal’s vector components to contour C. The plastic area’s contour L surrounds entirely the hole C. See Fig. 1.

Fig. 1. Elastic-plastic border near the hole C

In this case on contour C, apart from the condition (16), also fulfilled is the yield condition (7). Thus on C there are two conditions:

l1 T1 + l2T2 = Tn = Tn, a13 (t 1 )2 + Й23 (t2)2 = 1. From the conditions (17) we find the stress components on contour C:

1 l2 2 1 Tn 2 a1312Tn T l1 л/li a23 + l2 a13 — a13a23тП

T = -y- T + —, T = -------------------~2------------------------.

l1 l1 l2 a23 + l2 a13

From this point on, to be definite, in formulas (18) we will be selecting the upper sign.

(17)

(18)

3. The use of conservation laws to find stress components in the area

Assume the point M (xm ,ym) lies beyond the contour C. Let us draw a circumference with radius £ with the centre at the point M. We have £ : (x — xm )2 + (y — ym )2 = £2. Assume D is

a line connecting the point M with the contour C. We obtain a closed contour consisting of the circumference e, the segmant P and the contour C. See Fig. 2.

Fig. 2. Calculating the contour integral around the singular point M

From (15) we obtain

Ady — Bdx + Ady — Bdx + Ady — Bdx + i Ady — Bdx = 0. (19)

Jo Jp+ Jp- Je

The sum of the second and the third summands in (19) is equal to zero, because the integrals are calculated in different directions. Finally from (19) we have

I Ady — Bdx = — j) Ady — Bdx.

(20)

Let us convert the right-hand part of equation (20) introducing parametrisation x = e cos t, y = e sin t, 0 < t < 2n. As a result we have

r2n

j) Ady — Bdx = e j (A cos t + B sin t) dt.

Assume in (15)

x2 + ny

1 = —

y

x2 + ny2

r2n

(21)

(22)

(23)

Then from (21) we obtain

n p2n p2П

Ф A1 dy — B1 dx = e (A1 cos t + B1 sin t) dt = т1 dt = 2пт1 (xm ,ym).

Je J 0 J 0

The last equality in (23) is obtained with the use of the mean-value theorem with e tending to zero.

Assume in (15)

A = v-y д1 = - - (24)

Ay в 1 i

x2 + ny2 ’

л/n x2 + ny2 '

r2n

Then from (21) we obtain

p р2п р'2П

® A2dy — B2dx = e (A2 cos t + B2 sin t) dt = т2dt = 2пт2 (xm,ym). (25)

e 0 0

The last equality in (25) is obtained with the use of the mean-value theorem with e tending to zero.

x

1

a

From formula (20), and also from (23) and (25) we obtain

/ Aidy - Bidx = -2пт1 (xm, ym), / A2dy - B2dx = -2пт2 (xm, ym). (26)

jc Jc

Conclusion

Formulas (26) offer the opportunity to find stress components in any point xm, ym beyond the contour C. This allows us to determine the boundary between the elastic and plastic areas. If the plasticity condition is met а13(т1)2 + а23(т2)2 = 1 at the point xm,ym then this point belongs to the plastic area, if in the point the condition а13(т1 )2 + а23(т2)2 < 1 is met, then to the elastic area.

Note. The formulas found above allow us to solve elastoplastic problems even if the plastic contour does not entirely surrounds the contour C, provided that on the contour C the plasticity condition (7) is fulfilled.

References

[1] B.D.Annin, G.P.Cherepanov, Elastic-plastic problem, Novosibirsk, Nauka, 1983 (in Russian).

[2] S.G.Lehnitsky, Theory of elasticity of an anisotropic body, Moscow, Nauka, 1977 (in Russian).

[3] S.I.Senashov, A.M.Vinogradov, Proc. Edinburg Math. Soc., 31(1988), no. 3, 415-439.

DOI: 10.1017/S0013091500006817

[4] P.P.Kiryakov, S.I.Senashov, A.N.Yakhno, Application of symmetries and conservation laws to solving differential equations, Novosibirsk, Ros. Acad. nauk. Sib. otd., 2001 (in Russian).

[5] S.I.Senashov, O.V.Gomonova, A.N.Yakhno, Mathematical problems of two-dimensional equations of ideal plasticity, Krasnoyarsk, Izd. SibGAU, 2012 (in Russian).

Анизотропная антиплоская упругопластическая задача

Сергей И. Сенашов Ирина Л. Савостьянова

Сибирский государственный университет науки и технологий им. М. Ф. Решетнева

Красноярск, Российская Федерация

Ольга Н. Черепанова

Сибирский федеральный университет Красноярск, Российская Федерация

Аннотация. В работе решена анизотропная антиплоская упругопластическая задача о напряженном состоянии в теле, ослабленном отверстием, ограниченном кусочно-гладким контуром. В статье приведены законы сохранения, которые позволили свести вычисления компонент тензора напряжений к криволинейному интегралу по контуру отверстия. Законы сохранения дали возможность найти границу между упругой и пластической областями.

Ключевые слова: анизотропная упругопластическая задача, антиплоское напряженное состояние, законы сохранения.

Journal of Siberian Federal University. Mathematics & Physics 2020, 13(2), 218-230

DOI: 10.17516/1997-1397-2020-13-2-218-230 УДК 517.9

Asymptotic Analysis of Retrial Queueing System M/M/1 with Impatient Customers, Collisions and Unreliable Server

Elena Yu. Danilyuk* Svetlana P. Moiseeva^

National Research Tomsk State University Tomsk, Russian Federation

Janos Sztrik

University of Debrecen Debrecen, Hungary

Received 29.11.2019, received in revised form 04.12.2019, accepted 20.01.2020 Abstract. The retrial queueing system of M/M/1 type with Poisson flow of arrivals, impatient customers, collisions and unreliable service device is considered in the paper. The novelty of our contribution is the inclusion of breakdowns and repairs of the service into our previous study to make the problem more realistic and hence more complicated. Retrial time of customers in the orbit, service time, impatience time of customers in the orbit, server lifetime (depending on whether it is idle or busy) and server recovery time are supposed to be exponentially distributed. An asymptotic analysis method is used to find the stationary distribution of the number of customers in the orbit. The heavy load of the system and long time patience of customers in the orbit are proposed as asymptotic conditions. Theorem about the Gaussian form of the asymptotic probability distribution of the number of customers in the orbit is formulated and proved. Numerical examples are given to show the accuracy and the area of feasibility of the proposed method.

Keywords: retrial queue, impatient customers, collisions, unreliable server, asymptotic analysis.

Citation: E.Yu.Danilyuk, S.P.Moiseeva, J.Sztrik, Asymptotic Analysis of Retrial Queueing System M/M/1 with Impatient Customers, Collisions and Unreliable Server, J. Sib. Fed. Univ. Math. Phys., 2020, 13(2), 218-230. DOI: 10.17516/1997-1397-2020-13-2-218-230.

The ever increasing volume of information for designing communication systems in an optimal way requires new methods and approaches. More and more business processes involve big data transmission under limited capacities of devices. Therefore, developing of appropriate mathematical models of modern telecommunication systems and modifying of existing ones are very important. Queueing systems with repeated calls, or retrial queueing systems are suitable models for telecommunication systems. They are characterized by the feature that an arriving customer finding the server busy does not join a queue and does not leave the system immediately, but goes to some virtual place (orbit), and then it tries to get service again after some random time. A comprehensive description and comparison of classical queueing systems of retrial queues can be found in books by J.Artalejo and A. Gomez-Corral [1], J. Artalejo and G. Falin [2], G.Falin and J. Templeton [3], just to mentions some of them.

* daniluc.elena.yu@gmail.com https://orcid.org/0000-0002-7016-492X tsmoiseeva@mail.ru https://orcid.org/0000-0001-9285-1555

tsztrik.janos@inf.unideb.hu https://orcid.org/0000-0002-5303-818X

© Siberian Federal University. All rights reserved

The present paper generalizes the results obtained in [4,5]. We find the asymptotic stationary distribution of the number of calls in the orbit for the system under consideration. Collisions in the model usually arise in the analysis of communication networks when another message is transmitted during the transmission of a previous message. Such messages collide. They are considered distorted and both go into the orbit from where they ask the device for servicing again after a random delay time, see for example [4-8].

Impatience of calls in the orbit is understood as the case when a customer in the orbit can leave the orbit after a random time without service [4,5,7,9-11]. But there is another way to specify impatience, for example, non-persistence, balking and reneging are used [8,12-15]. Balking and reneging are fundamental concepts in queuing introduced by Anker, Gafarian [16], Haight [17] and Bareer [18]. They state that an arriving customer shows the least interest in joining a system which is already crowded. This behaviour is referred to as balking. Balking was applied to retention of reneged customers [19-23]. A comprehensive review of queueing systems with impatient customers can be found in [24].

In practice some components of the systems are subject to random breakdowns. Then it is very important to study reliability of retrial queues with server breakdowns and repairs because of the limited ability of repairs and heavy influence of the breakdowns on the performance of the system. Retrial queues with an unreliable server were studied , see for example [5,14,25,26] and references therein.

More references on important papers devoted to the research of retrial queueing systems of various types (with impatient customers, collisions, and unreliable server) are given in our previous papers [4,5,7,9-11,25,26].

The novelty of our contribution is the inclusion of breakdowns and repairs of the service into our previously developed models to make the problem more realistic and hence more complicated. In the present paper we continue to use an asymptotic analysis method developed at the Tomsk State University that is widely applied for the study of RQ -systems. This method makes it possible to obtain analytical result for different types of queueing systems and networks under specific asymptotic conditions.

The structure of the present work is as follows. Mathematical model of the novel retrial queueing system discussed in the paper and the problem statement are presented in Sect. 1. In Sect. 2 the detailed description of the model and the system of Kolmogorov equations for the stationary state probabilities are given. Sect. 3 consists of the solution of the problem by the asymptotic analysis method. Theorem on stationary probability distribution of the number in the orbit for retrial queueing system of M/M/1 type with impatient calls in the orbit, collisions and unreliable server under a long delay of calls in orbit and long time patience of calls in the orbit condition is formulated and proved in this section. Sect. 4 deals with some numerical examples that prove the theoretical results and illustrate the applicability of the proposed approach. Sect. 5 concludes the paper.

1. Description of the mathematical model

We consider a single server RQ-system with Poisson arrival process with parameter A for the primary calls. A customer that finds the server idle takes it for service for an exponentially distributed random time with parameter ц. If the server is busy an arriving customer (either from the source or from the orbit) enters into a "collision" and both go into orbit. In the orbit each customer, called secondary calls, independently of others waits for a random time.

The waiting time is exponentially distributed with parameter a. If the server is busy again the request tries to occupy the device to obtain servicing as soon as possible. If the server is idle the secondary customer occupies it for service for an exponentially distributed random time with parameter ц, that is, no difference between the service of primary and secondary calls.

We assume that server is unreliable, that is, the lifetime is supposed to be exponentially distributed with rate 70 if the server is idle and with parameter 71 if it is busy. When the server breaks down it is immediately sent for repair and the recovery time is assumed to be exponentially distributed with rate 72. When the server is down the primary sources continue generation of customers and send them to the server. Similarly, customers may retry from the orbit to the server but all arriving customers immediately go into the orbit. Furthermore, in this unreliable model we suppose that interrupted request goes to the orbit immediately and it's next service is independent of the interrupted one. All random variables involved in the model construction are assumed to be independent of each other.

Moreover, a customer in the orbit leaves the system without service after a random time which has an exponential distribution with rate a, demonstrating the "impatience" property. Fig. 1 shows the model of the RQ-system M/M/1 with impatient customers, collisions and unreliable server.

Our aims is to find the stationary distribution of the number of customers in the orbit for the described system.

Fig. 1. Retrial queue M/M/1 with impatient customers in the orbit, collisions and unreliable server

2. System of the Kolmogorov differential equations

Let us consider the Markov process {k(t), i(t)} of changing the states of the RQ-system under consideration, where i(t) is the number of customers in the orbit at time t, i(t) =0,1, 2,... and k(t) defines the state of the server at time t and takes one of three values:

{0, if the device is idle,

1, if the device is busy,

2, if the device is down (under repair).

The joint probability that device is in state k at time t and i customers are in orbit is denoted

by P {k(t) = k, i(t) = i} = P(k, i,t).

To obtain the probability distribution P {k(t) = k, i(t) = i} = P(k, i,t) for the states of the

considered RQ-system we derive a system of the Kolmogorov differential equations ' dP (0, i, t)

dt

dP (l,i,t) dt

dP (2,i,t) dt

— (A + ia + ia + Yo) P (0, i, t) + (i + 1) aP (0, i + 1, t) +

+ (i — 1) aP (1,i — 1, t) + fiP (1, i,t) + AP (1,i — 2, t) + Y2P (2,i,t). = — (A + л + ia + ia + Yi) P (1, i, t) + AP (0, i, t) +

+ (i + 1) aP (0,i + 1, t) + (i + 1) aP (1, i + 1, t),

= — (A + ia + Y2) P (2, i, t) + AP (2, i — 1, t) +

+ (i + 1) aP (2, i + 1, t) + YoP (0,i,t) + YiP (1, i — 1, t),

(1)

i = 0,1, 2,....

Since customers are "impatient" the considered system has a stationary distribution for any values of A and /л. Let lim P (k,i,t) = P (k,i) = Pk (i), к = {0;1;2}. Then we can write system (1) in the following form

— (A + ia + ia + Yo) Po (i) + (i +1) aPo (i + 1) + (i — 1) aPi (i — 1) + jiPi (i) +

+ APi (i — 2) + Y2P2 (i) = 0,

— (A + /л + ia + ia + Yi) Pi (i) + APo (i) + (i + 1) aPo (i + 1) + (i + 1) aPi (i + 1) = 0,

(2)

— (A + ia + Y2) P2 (i) + AP2 (i — 1) + (i + 1) aP2 (i + 1) + YoPo (i) + YiPi (i — 1) = 0,

TO

E(Po (i)+ Pi (i)+ P2 (i)) = 1,

i=o

i = 0,1, 2,....

System (2) is a system of difference equations of infinite dimension with variable coefficients. Such system is very difficult to solve. Therefore, to get solution we propose two approaches: asymptotic and numerical ones.

The numerical algorithm for finding the stationary probabilities is based on the reduction of dimension of system (2). To do that we represent system (2) for i = 0,1,2,... ,N as

PS = B,

(3)

where the row vector P of dimension 3 (N +1) is the desired stationary probability distribution of the number of customers in the orbit for each state of the device k = {0, 1, 2}

P =(P(0) P(1) P(2))

and P(0), P(1), P(2) are row vectors with elements P(0, i), P(1, i), P(2, i), i = 0,1, 2,..., N, respectively. Matrix S of dimension 3(N + 1) x (3(N + 1) + 1) is represented in the block form as

where Su = ||sii||f+i, Si2 =

( Sii Si2 Si3 Si4\

S = 1 S2i S22 S23 S241,

\S3i S32 S33 S34

||sij2||f +\ Si3 = I^H N+i a 1 , S21

- Il,2i|\N+i

s

ij Mi

, S22 =

= lls22||N+i

s

ij || 1

S23 = +i, S3i = ||s3j1||f+i, S32 = ||s32||f+i, S33 = ||s33||N+i are sparse matrices with

non-zero elements defined as

s\} = -A - {i - l)(a + a) - Yo, s1+M = ia,

sii №, si,i+1 (i 1)a, si,i+2 A,

sii 72,

sii = A sM-1 = a

sii = -(A + № + 71 + (i - l)(a + a)), s2++1!i = ia,

sii 70,

23

si,i+1 71,

sii = -(A + Y2 + (i - 1)a) s3++1,i = ia, s33+1 = A-

Blocks 514, S24, S34 are unit vector columns of dimension (N + 1), row vector B = ||6j|| of dimension 3(N + 1) + 1 is a row of free coefficients with elements bn =0 (n = 0,1,2,...,N - 1), bN = 1.

We solve (3) with the use of the Mathcad software package. We choose N to be so large that probabilities P(0,N), P(1,N), P(2, N) are equal to the machine zero.

The numerical algorithm provides satisfactory accuracy but it has a drawback due to the limited computational capacity of the computer. Therefore, analytical methods are needed to calculate the probability distribution of the number of customers in orbit for the considered RQ-system with impatient customers, collisions and unreliable server. They allow us to find a distribution for a system of any dimension. Thus the alternative to the numerical method is the method of asymptotic analysis.

3. Asymptotic analysis

To find the solution of system of equations (2) we propose another approach by using the method of asymptotic analysis under the assumption that there is a long delay between customers from the orbit and high "patience" customers, i.e., when a ^ 0, a ^ 0. We summarize the results of our study in the next Theorem 3.1.

Theorem 3.1. The stationary distribution of the number of customers in the orbit in the RQ-system M/M/1 with impatient customers, collisions and unreliable server under the above assumptions and conditions a = qa, q > 0 is the asymptotically normal distribution with mean G1/a and variance G2/a. Here

G1

A - рН,1 q ''

G2

G1 (q + До) + Y1/0 + AR1 Ro - Д1 + q

/1 =

(2A + (1 - q) G1) Д1 - (1 + q) G1Ro

(4)

(5)

(6)

Yo + Y1 + 2Y2

R1 is the probability that the server is busy in the stationary regime of system operation. It is determined by equation

(Yo + Y1 + 2Y2) rR2i - СД1 + A (1 + q) Y2 =0, Д1 £ [0; 1].

(7)

c = (1 + q) A (70 + 71 + 272) + q (70 + 72) (m + Yi) + MY2.

R0 is the probability that the server is idle in the stationary regime of system operation. It is determined by equation

R0

Y2 ~ (Yi + Y2) Ri Yo + Y2

Ro & [0;1].

(8)

Proof. The method of asymptotic analysis in queueing theory is the method of study of the equations to determine some characteristics of an queueing system under some limit (asymptotic) condition which is specific for any model and problem under consideration.

We introduce the partial characteristic functions as follows

Hk(u) = Y, ejuiPk (i), Hk(0) = £ Pk (i) 4 R-,

i=0 i=0

(9)

where j = %/—T, к = {0,1, 2}, and Rk are stationary state probabilities of the process k(t). It is obvious that H(u) = H0(u) + H1(u) + H2(u).

dHk (u)

Using (9) and Hk(u) =-----------= j У iejuiPk(i), к = {0,1, 2}, we can write system (2) as

du i=0

follows

— (A + 70) Hd(u) + j (a + a) H0(u) + pHi(u) — jae juH0(u) — jaejuH'i(u)+

+ Ae2juHi(u)+ Y2H2(u)=0,

— (A + m + Yi) Hi(u) + j (a + a) H[(u) + AH0(u) — jae-juHi(u) — jae-juH0(u) = 0,

— (A+72) H2(u) + jaH2(u) + AejuH2(u) — jae-juЩ(u) + 10H0W + ne?uHi(u) = 0.

(10)

Adding the first and the second equations by the third one of (10) we get the system below

' — (A + 70) H0(u) + (m + Ae2ju) Hi(u) + Y2H2(u) + j (a + a — ae ju) H0(u) —

— jaejuH'i (u) = 0,

AH0(u) — (A + m + Yi) Hi(u) — jae-juH0(u) + j (a + a — ae-ju^ H[(u) = 0, 70H0H + YiejuHi(u) — (A + 72 — AeJu) H2(u) + ja (1 — e-ju) Щ(u) = 0,

[A (eju + 1) + 7i] Hi(u) + AH2(u) + je-ju (a + a) H0(u) + j (ae-ju — a) H'i(u)+ + jae-juH2 (u) = 0.

(11)

System (11) is the basic system for further analysis of retrial queueing system of M/M/1 type with impatient customers in the orbit, collisions and unreliable server under a long delay of customers in orbit (a ^ 0) and long time patience of customers in the orbit (a ^ 0) conditions.

The proof of Theorem 3.1 is carried out in two stages.

Stage 1. Finding the first-order asymptotic.

Let us make the substitutions a = e, a = qe, u = ew, Hk(u) = = Fk(w,e), к = {0,1, 2} in basic system (11), where e ^ 0.

Since H'k (u) = -

1 dFk(w,e)

, k = {0,1,2} systems of equations (11) can be written as

£ dw

- (X + yo) F0(w,£) + (л + Xe2jwe) Fi(w,£) + Y2F2(w,£)+

dFi(w, £)

+ j (1+ q - <,e-‘«) - je^

dw

XFo(w, £)-(X + л + Yi) Fi(w, £) -je-jwe

dw

dFo(w,£)

dw

0,

+ j (1+q-qe~jw) dFwH=^

dw

YoFo(w, £)+YiejwEF1 (w, £) - (X+Y2 -Xejwe) F2(w, £) +jq (l-e-jwe) dF'2(w,£') = 0,

(12)

dw

[X (ejwE + У + yi] Fi(w, £) + XF2(w, £) + je jwe (1 + q)

dFo(w,£) dw

+ j (qe-jw - 1)

dFi(w, £) dw

+ jqe

-jwe

dF2(w, £) dw

0.

The transformation of equations (12) under £ ^ 0 with Fk(w) = lim Fk(w,£), Fk(w)

e—^ 0

dFk (w) dw

, k = {0,1, 2} leads to

' - (X + yo) Fo(w) + (л + X) F1(w) + Y2F2(w) + jF0(w) - jF[(w) = 0, XFo(w) - (X + л + Yi) Fi(w) - jFQ(w) + jF[(w) = 0,

YoFo(w) + YiFi(w) - Y2F2(w) = 0,

У [Fo(w) + Fi(w) + F2(w)] - лFl(w) + jq [Fq(w) + F{(w) + F2(w)] = 0. We seek solution of equation (13) Fk (w), k = {0,1,2} in the form

(13)

Fk(w) = Rk$(w), k = {0,1, 2} , (14)

where R0, Ri, R2 are defined in (9), R0 + Ri + R2 = 1, Rk = Hk(0) = Fk(0), k = {0,1, 2}, and

ФУ) is an unknown function.

Substituting (14) into (13), we obtain the system of differential equations with respect to function ФУ)

[- (X + yo) Ro + (л + X) Ri + Y2R2] &(w) = j (Ri - Ro)

dФ(w)

dw

[xro - (X + Л + Yi) Ri] ф(ш) = j (Ri - ro) YoRo&(w) + YiR^(w) - Y2R2Ф(w) = 0,

dф(w)

dw

(15)

[X (Ro + Ri + R2) - лRl] Ф^) — -jq [Ro + Ri + R2]

According to equations (15), we can find

dФ(w)

dw

Ф^) = exp {Gijw} , (16)

where Gi is given in (4). It follows from the forth equation (15).

It is obvious that solution of system (15) exists when the following equalities are satisfied

Г (X + Yo) Ro - (л + X) Ri - Y2R2 = XRo - (X + л + Yi) Ri,

1 Y0R0 + YiRi - Y2R2 =0, (17)

1 Ro + Ri + R2 = 1.

Expressions for Ro,R1,R2 £ [0; 1] can be obtained from system of equations (17). They are

D 72 - (yi + 72) Ri d 7o - (70 - 71) Ri

Ro —-----------;-------, R2 —----------■--------,

70 + 72

(18)

70 + 71

and R1 is the root of equation

(yo + 71 + 272) I^R2i - cRi + X (1 + q) 72 — 0, (19)

c — (1 + q) X (70 + 71 + 272) + q (yo + 72) (l + 71) + ly2-

Equation (19) has at least one root R1 £ [0; 1], and the proof of existence of R1 is similar to that in [4].

Using (14), (16) and e — a we can write the expression for the partial characteristic functions as follows

Hk(u) — Fk(w,e) — Fk(w) + o(e) « Fk(w) — Fk (— Rk exp j —

(20)

k — {0,1,2}, and Ro, R1,R2 £ [0; 1] are defined in (18), (19).

Using (20) and Ro + R1 + R2 — 1 the pre-limit characteristic function H(u) — H0(u) + H1(u)+ +H2(u) under the assumption of a long delay of customers in the orbit and their high "patience" can be approximated by function h1(u)

Mu) — exp <J — ju\ ,

(21)

which is the first-order asymptotic characteristic function (or the first-order asymptotic). Stage 2. Finding the second-order asymptotic.

Taking into account (21), we assume

Hk(u) — exp <{ —M [ H(2)(u), k — {0,1, 2} ,

(22)

in basic system of equations (11). Then systems of equations (11) can be rewritten as follows

—

X + 70 +-----(a + a — ae ju)

H02)(u) + M + Xe2ju + GMu) H12)(u)+

(2) {

+ 72H2(2)(u) + j (a + a - ae ju^F±F

dH02)(u)

du

jae

M2)(u

du

(X + G1e-ju) H02)(u) -

-ju dH0

X + Lb + 71 + — (a + a — ae ju\ a

(2)

H(2)(u)-

- jae

7oH02) (u) + 71ejuH1z> (u) -

* dH02,'“»+ j (a + a - ae-ju) dH±M — 0,

du

(2)

du

X + 72 - Xeju + Gaa (1 - e-ju)

H2(2)(u)+

+ ja (1 - e ju)

dH22)(u)

du

G

-e-ju—1 (a+a) H02)(u) + a

X(eju + ^+71-----1 (ae-ju -a)

H12)(u)+ (X- —aae-u )x

(2), „• dH02)(u) ( ) dHF1 (u) dHF* (u)

x H22)(u)+je-ju (a+a)----+j (ae-ju - a)--------+ jae-ju 2-^

2 du du du

(2)

(2)

0

0

Let a = e2,a = qe2, u = ew, (u) = F(2^(w, e), к = {0,1, 2}, where e ^ 0. Then system

(23) after some transformations becomes

' - [A + 70 + Gi (1 + q - qe-jWE)] f(2) (w, e) + (ц + Ae2jWE + GiejWE) F(2\w, e) +

+ 72F2,2)(w,e) + je (1 + q - qe jWE)

A2) /

dF(2)(w,e)

dw

jee

(A + Gie-jwe) F0(2) (w, e) - [A + ц + 71 + Gi (l + q - qe-jWE)] F(2)(w, e)-

. dF(2)(w, e)

dw

(2)

- jee

p(2),

-jwe dF02)(w,e) , (, , q qe-jws) dFl2) (w,e)

dw

+ je (1 + q - qe-jwE)

dw

7oF(2)(w,e) + 71 ejWEF(2)(w,e) - [A + 72 - AejWE + Giq (1 - e-jWE)} F(2)(w,e) +

(2) i

+ jqe (1 - e-jWE)

dF(2)(w,e)

dw

- e-jWEGi (1 + q) F(2)(w, e) + [A (ejWE + 1) + 71 - Gi (qe-jWE - 1)] F(2)(w,e)+

(2)

+ (A - Giqe-jWE) F(2)(w,e)+ jee-jWE (1 + q)

dF(2\w,e)

dw

+

+ je (qe-jWE - 1)

dF(2)(w,e)

dw

dF22)(w,e)

dw

+ jqee-

0.

When e ^ 0 in (24) and lim F(\w, e) = F(2) (w), к = {0,1, 2} we obtain

(24)

- (A + 70 + Gi) f0 \w) + (ц + A + Gi) F-f \w) + 72Ff \w) — 0

(2)

(2)

(A + Gi) Fo (w) - (A + ц + 7i + Gi) F( ^(w) — 0,

(2)

7oFo2\w) + 7iF^>(w) - 72F^>(w) = 0. (2)

(2)

(2),

- Gi (1 + q) F0Z) (w) + [2A + 71 - Gi (q - 1)] F{2)(w) + (A - Giq) F^> (w) = 0.

The solution of systems of equations (24) has the following form

(f(2) (w,e) = (Rk + jwefk) Ф{2) (w) + o (e2) , к = {0,1, 2} ,

1 Ro + Ri + R2 = 1,

(2)

(25)

(26)

where R0, R1, R2 are defined above, f0, f1,f2 are constants and function Ф(2)^) is to be determined.

Substituting (26) into (24) and taking into account (25), with the proviso that e ^ 0 one can write the system as

[(A + 70 + Gi) f0 + GiqR0 - (ц + A + Gi) fi - (2A + Gi) Ri - 72f2] w$(2')(w) =

dM2)(w)

( R0 - Ri )

dw

[(A+Gi) f0-G1R0-(A+ц+71 +Gi) fi - GiqRi] w<^(2\w) = R - Ri)

d^^2^ (w)

dw

[70 f0 + 7ifi + 7iRi - 72 f2 - (Giq - A) R2] w§(2) (w) = 0,

[(2A + 71 + Gi - Giq) fi + (A + Giq) Ri - (A - Giq) f2 + GiqR2 - Gi (1 + q) f0 +

d$>(2) (w)

+ Gi (1 + q) R0] w4(2)(w] = [(1 - q) Ri - (1 + q) R0 - qR2]

dw

0

0

0

where R0, Ri and Gi are defined in (18), (19) and (16), respectively. The solution of system (27) has the form

Ф(2) (w)

exp

G2

(jw)2 2

(28)

where G2 is defined in (5).

Using the same transformation as for the first-order asymptotic and additional conditions f2 — /i — f0 = 0 and fi — f0 = 0, we finally obtain expressions for the solution of system (27)

(A + Yo + Gi) fo + GiqRo — (p + A + Gi) fi — (2A + Gi) Ri — Y2f2 =

= (A + Gi) fo — GiRo — (A + M + Yi + Gi) fi — GiqRii < Yofo + Yifi + YiRi — Y2f2 — (Giq — A) R2 = ° (29)

2fi + f2 =0, fi — fo = 0.

Making the reverse substitutions we obtain

H(2) («) = A2) (w,e)

(Rfc + jwefk) exp

G2

(jw)2

2

+ о (e2) « Rfc exp

G2 (ju)2 a 2

(30)

Then using (30), expressions (22) can be written as

Hk (u) = exp { fOl ju I Hk2\u) « Rfc exp i G ju +

G2 (ju) a 2

* = {0,1, 2} . (31)

Taking into account (31), characteristic function H(u) = Ho(u) + Hi(u) + H2(u). Assuming that customers in the orbit have long delays and the "patience" is high, we can see that distribution is the Gaussian one. Hence

Theorem 3.1 is proved.

h2(u) = exp

Gi . , G2 (ju)2

—Ju +------x-

a a 2

(32)

□

4. Numerical results

In this section we give some comments to Theorem 3.1 and several numerical examples are considered.

We construct asymptotic distributions of the probabilities of the number of customers in the orbit with parameters m =1, yo = 0.1, y1 = 0.2, y2 = 1, a = 2a for various values of the delay parameter a and parameter A. Then these distributions are compared with pre-limit (numerical) distributions obtained by the matrix method.

Fig. 2 shows one of samples for A = 0.7 and a = 0.01 (left picture) and a = 0.001 (right picture).

As a measure of proximity of two distributions the Kolmogorov distance

Д

max

O^i^N

E

k=0

Pm

i

(i) ^ ^ Pasimpt (i)

k=0

is used, where Pmatrix (i) is the probability distribution of the number of customers in the orbit obtained by the matrix method, Pasimpt (i) is the asymptotic probability distribution of the number of customers in the orbit.

Fig. 2. Asymptotic (dashed line) and pre-limit(numerical) (solid line) probability distributions of the number of calls in the orbit

Table 1. Values of the Kolmogorov distance

X/p Kolmogorov distance A

a = 0.1 a = 0.05 a = 0.01 a = 0.005 a = 0.001

0.5 0.161 0.101 0.023 0.016 0.009

0.7 0.117 0.066 0.020 0.016 0.013

0.9 0.092 0.048 0.021 0.020 0.018

1.1 0.075 0.039 0.024 0.023 0.022

1.5 0.055 0.035 0.030 0.029 0.029

2.0 0.046 0.037 0.035 0.035 0.035

Conclusion

Retrial queueing system of M/M/1 type with impatient customers in the orbit, collisions and unreliable server is considered in the paper. It is proved that stationary probability distribution of the number of customers in the orbit can be approximated by the Gaussian distribution under conditions of a long delay and a long patience time of the customers in the orbit. The accuracy of the approximation was compared with numerical results obtained with the use of the matrix analytic method.

The study was funded by Russian Foundation for Basic Research and Tomsk region (project no. 19-41-703002).

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Асимптотический анализ системы массового обслуживания с повторными вызовами M/M/1 с нетерпеливыми заявками, конфликтами и ненадежным прибором

Елена Ю. Данилюк Светлана П. Моисеева

Национальный исследовательский Томский государственный университет

Томск, Российская Федерация

Янош Стрик

Университет Дебрецена Дебрецен, Венгрия

Аннотация. В настоящей статье мы рассматриваем систему массового обслуживания с повторными вызовами (RQ-систему) типа Ы/М/1 с пуассоновским потоком поступающих в систему заявок и одним сервером, обслуживание которым имеет экспоненциальное распределение. Классическая модель RQ-системы усложнена наличием конфликтов заявок в системе, "нетерпеливых" заявок на орбите, а также "ненадежным" прибором, который выходит из строя и ремонтируется в функционирующей системе массового обслуживания. Время, через которое заявки с орбиты вновь обращаются к обслуживающему прибору; время, через которое заявки с орбиты покидают систему, время, в течение которого сервер находится в рабочем состоянии (в зависимости от того, занят прибор обслуживанием заявки или нет, а также время, в течение которого длится ремонт вышедшего из строя сервера, распределены экспоненциально. Мы используем метод асимптотического анализа для решения задачи нахождения распределения вероятностей числа заявок на орбите. В качестве асимптотического условия предлагается условие высокой загрузки системы и долгой "терпеливости" заявок на орбите. Формулируется и доказывается теорема об асимптотически гауссовском распределении вероятностей числа заявок на орбите. Приводятся численные результаты, демонстрирующие область применения полученных теоретических выводов.

Ключевые слова: RQ-система, нетерпеливые заявки, конфликты, ненадежный прибор, асимптотический анализ.