Heterogeneous System MMPP/GI(2)/∞ with Random Customers Capacities Текст научной статьи по специальности «Математика»

УДК 517.9

Heterogeneous System MMPP/GI(2)/to with Random Customers Capacities

Ekaterina V. Pankratova*

V. A. Trapeznikov Institute of Control Sciences of RAS Profsouznaya, 65, Moscow, 117342

Russia

Svetlana P. Moiseeva^

National Research Tomsk State University Lenina Ave., 36, Tomsk, 634050 Russia

Mais P. Farhadov*

V. A. Trapeznikov Institute of Control Sciences of RAS Profsouznaya, 65, Moscow, 117342

Russia

Alexandr N. Moiseev§

National Research Tomsk State University Lenina Ave., 36, Tomsk, 634050 Russia

Received 28.11.2018, received in revised form 14.01.2019, accepted 20.02.2019 A heterogeneous queuing system with an infinite number of servers is considered in this paper. Customers arrive in the system according to a Markov Modulated Poisson Process. The type of incoming customer is defined as i-type with probability pi (i = 1, 2). Each customer carries a random quantity of work (capacity of the customer). In this study service time does not depend on the customers capacities. It is shown that the joint probability distribution of the customers number and total capacities in the system is multidimensional Gaussian distribution under the asymptotic condition of an infinitely growing service time. .Simulation results allow us to determine an applicability area of the asymptotic result.

Keywords: infinite-server queueing system, random capacity of customers, Markov Modulated Poisson Process.

DOI: 10.17516/1997-1397-2019-12-2-231-239.

Queueing systems represent a powerful mathematical tool for investigating the performance of a wide variety of real-life systems ranging from telecommunication networks to financial markets, from computer architectures to supply chain management and airplane traffic control. Analytical tractability of the corresponding models strongly depends on the nature of the underlying processes: mathematical model of arrival process, discipline of service, and on the system structure. A new trend in the study of resource queueing systems is the analysis of systems with non-Poisson arrivals and non-exponential service time. Although physical resources are always finite, quite often it is easier to study queueing systems in which it is assumed that the corresponding parameters can reach infinite values. The fundamentals of the theory of queueing systems for random-capacity customers can be found in [13,14]. The single-server queueing systems for random-capacity customers with Poisson input flow were considered [15] where the servicing time is distributed exponentially and arbitrarily under the assumption that the customer capacity and

* [email protected]

t [email protected]

^ [email protected]

§ [email protected] © Siberian Federal University. All rights reserved

the servicing time are independent, and the AQM mechanism is realized in them. Similar results were established for the infinite-server system with exponentially and arbitrary distributed servicing time [2,9-11]. We note that the queueing system with random-capacity customers, processor sharing, and limited capacity of memory using specific algorithm was discussed in detail [1]. In this work we consider an infinite-server queueing system fed by non-Poisson arrivals with random customers capacities. Queues with random customers capacities are useful for analysis and design issues in high-performance computer and communication systems, in which service time and customer volume are independent quantities. For instance, performance analysis of LTE (Long Term Evolution) networks was carried out in terms of flow level dynamics and the amount of required radio resources does not depend on the duration of the flow [7]. Such queues are also important in modeling devices, where it is necessary to calculate a sufficient volume of buffer for data storing [7,12]. A new trend in the study of queueing systems is the analysis of the systems with non-Poisson arrivals and non-exponential service time. The main contribution of this paper is to extend such analysis, focusing on the properties of the multidimensional process of the number of customers and the total capacity in the system when an infinite-server queue is fed by MMPP arrivals with random capacities and non-exponential service time distribution.

1. Matematical model

Let us consider the queueing system with unlimited number of servers of two different types [10,11]. Each customer carries a random quantity of work (capacity of the customer) [2] (Fig.1).

Customers arrive in the system according to a Markov Modulated Poisson Process (MMPP). The input process is defined by its generator matrix Q = \\qij|| of size K x K and by conditional rates Ai,..., AK typically composed into the diagonal matrix A = diag{Ai,..., AK}. Let us denote the underlying Markov chain of the MMPP by k(t) G 1,2,... ,K. At the time of occurrence of the event in the MMPP-flow only one customer flows in the system. The type of incoming customer is defined as ¿-type with probability pi (i = 1, 2). It goes to the appropriate device type, where its service is performed during a random time with distribution function Bi(x) (i = 1,2), according to the type of the customer. Let us assume that each customer of i-type has some random capacity vi > 0 (i = 1,2) with distribution function Gi(y) (i = 1, 2).

Fig. 1. Heterogeneous queue MMPP/GI/ to with random customers capacities

Denote the number of each type's customers in the system at time t by {i1 (t), i2(t)} and their total capacity by {Vi(t),V2(t)}. Four-dimensional stochastic process {i1(t),i2(t),V1(t),V2(t)} is the goal of the study. This process is not Markovian, therefore, we use the dynamic screening method [6] for its investigation.

Let the system be empty at moment t0, and let us fix some arbitrary moment T in the future. Consider three time axes that are numbered from 0 to 2 (Fig.2). Let axis 0 shows the epochs of customers' arrivals, while axes 1 and 2 correspond to two-dimensional screened process. Let S1(t) be a probability that the arriving customer generates a point on the first axis of the screened process and S2(t) be a probability that it generates a point on the second axis. The customer

does not generate any points with probability 1 — Si(t) — S2(t). Probability Si(t) is equal to the probability that a customer of ¿-type (i = 1, 2) arriving at time t is serviced in the system at the moment T, Si(t) = 1 — Bi(T — t), i = 1, 2, for t0 < t < T.

Fig. 2. Screening of the customers arrivals

Let us denote the number of arrivals screened before the moment t and their total capacity by mi(t), m2(t) and wi(t), w2(t),respectively. As it was shown [5], the probability distribution of the number of customers in the system at the moment T coincides with the probability distribution of the number of screened arrivals on the axis

P{¿i(T) = mi, i2(T) = m2} = P{mi(T) = mi,m2(T) = m2]

for all m = 0,1, 2,...

It is easy to prove the same property for the extended process {ii(t), i2(t), Vi(t), V2(t)}:

P{ii(T) = mi,i2(T) = m2,Vi(T) < wiV(T) < W2} = = P{mi(T) = mi, m2(T) = m2, wi(T) < wi, W2(T) < W2}

for m = 0,1, 2,... and 2 ^ 0.

2. Kolmogorov differential equations

Let us consider the five-dimensional Markovian process {k(t), mi(t), m2(t), wi(t), w2(t)}. The probability distribution of this process is

P (k, mi, m2, wi, w2,t) = P {k (t) = k, mi (t) = mi, m2 (t) = m2, wi (t) < wi, w2 (t) < w2} .

Taking into account the formula of total probability, we can write the following system of Kolmogorov differential equations

dP(k, mi, m2, wi, w2, t) dt

^k [(pi (1 — Si (t)) + p2 (1 — S2(t)) — 1) P (k, mi,m2, wi,w2,t)+

fWl

+PiSi (t) P (k,mi — 1,m2 ,wi — y,w2,t)dGi(y)+ J0

r W2

+P2S2 (t) P (k,mi ,m2 — 1,wi,w2 — y,t)dG2(y)] + y^ P (v,mi,m2,wi,w2,t)qvk

0

for k =1,...,K, mi = 0,1, 2,..., wi > 0, i = 1, 2. We introduce the partial characteristic function

TO TO

h(k,ui,u2,zi,z2,t)= J2 J2 ejUimiejU2m2 ejziWlejz2w2P(k,mi,m2,dwi,dw2,t),

0

mi =0 m2 = 0

where j = %/—1 is the imaginary unit. Then we can write the following equations ^(M!,^!,^) = ,z2,t)Xk [p1S1(t)(ejU1 G*1(z1) — 1)+

+P2S2(t)(eJu2G1(z2) — 1)] + ^ h(v,U1,U2,Z1,Z2,t)qvk

for k =1,...,K, where G1(zi) = f ejziy dG(y) i = 1, 2.

0

Let us rewrite this system in the matrix form

dh(u1,u2,z1,z2,t) ,, ,,

-= h(ui, U2, zi,Z2,t)

dt

with the initial condition

Y,Pi(ejUiG*(zi) - l)Si(t)A + Q

i=1

(1)

h(ui,U2,zi,z2,to) = r, (2)

where h(u, U2, z1, z2, t) = [h(1,U1,U2, z1, z2,t),..., h(K, U1,U2,z1, z2,t)]andr = [r(1),..., r(K)] represents the stationary distribution of the underlying Markov chain, i.e., vector r satisfies the following linear system

{ rQ = °' (3)

[re =1, w

where e is a column vector with all entries equal to 1.

In general, the exact solution of equation (1) is not available but it may be found subject to asymptotic conditions. We consider in the paper the case of infinitely growing service time.

3. Asymptotic analysis

We formulate and prove the following statements.

Theorem 3.1. The first-order asymptotic characteristic function of the probability distribution of the process {k(t),m1(t),m2(t),w1(t),w2(t)} has the form

h(1)(u1,u2,z1,z2,t) = exp <|jK1 ^pi(ui + ziaH)bi j ,

oo oo

where k1 = rAe, bi = f(1 — Bi(x))dx and a1i = J ydGi(y) is the mean customer capacity.

00

Proof. Let us perform substitutions

1 = £, t£ = t, Si(t) =Si (t), Ui = £Xi, zi = eyi, bi

h(u1,u2,z1,z2,t) = f1 (x1,X2,y1,y2,T,e). Then problem (1) takes the form

dfi(xi,x2 ,yi,V2,T,e) , .

£-dT-= ii(xi,X2,yi,y2,T,e)

Q + EPi(ej£XiG*(eyi) - 1) Si (t)A

i=i

, (4)

with the initial condition

fi(xi,X2,yi,y2,To ,e)= r. (5)

Let us find the asymptotic solution (where e ^ 0,) of problem (4)-(5). Let e ^ 0 in (5). Then we obtain

fi(xi,x2,yi,y2,T )Q = 0.

Comparing this equation with the first one in (3), we can conclude that fi(xi,x2,yi,y2,r) can be expressed as

fi (xi,x2,yi,y2,T) = r$i(xi,x2,yi,y2,r), (6)

where $i(xi,x2,yi,y2,T) is some scalar function that satisfies the condition

$i(xi,x2,yi ,y2,To) = 1.

Let us multiply (4) by vector e, substitute (6), divide the result by e and assume that e ^ 0. Then, taking into account (4), we obtain the following differential equation for $i(xi, x2,yi, y2,T)

d$i(xi ,x2,yi,y2,T) _ v^ s -—-= JKi$i (xi,x2,yi,y2,T Pi Si (t )(xi + yiaii).

i=i

Taking into account the initial condition, we obtain the solution of this equation

$i(xi,x2,yi,y2, t) = exp <| jKi ^ Pi(xi + yi au) J Si (z)dz^ .

Using substitutions, we can write first-order asymptotic characteristic function of the probability distribution of the process {k(t),mi(t),m2(t),wi(t),w2(t)}

h(i\ui,u2,zi,z2,t) = exp <|jKi ^pi(ui + ziaii)bi j .

The theorem is proved.

The main result of the paper is the following theorem.

Theorem 3.2. The second-order asymptotic characteristic function of the probability distribution of the process {k(t), mi(t), m2(t), wi(t), w2(t)} is

( 2 j 2 2 h(2) (ui ,u2,zi, z2,t) = exp < jKi^2 Pi (ui + ziaii)bi + — Ki ^ pi(u.2 + z2a2i)bi+

I i=i i=i

+j2KiJ2 Piuizia2ubi + j2 K2

2 2 2 2

Pi (ui + ziau)b2i + ^ PiPg(ui + ziau)(ug + zgaig)bibg

i=i i=i g=i,g=i

where k2 = d(A — KiI)e, a2i = f y2dGi(y), b2i = j(1 — Bi(x))2dx and the row vector d satisfies

00

the linear matrix system

( dQ = r(KiI — A), de = 0.

Proof. Let us represent the original characteristic function h(ui,u2, zi, z2,t) in the form

h(ui,u2,zi,z2,t) = h2(ui,u2,zi, z2,t) exp < jKi ^Pi(u + ziau) Si (z)dz> ,

{ i=i Jt<0 J

where h2(ui,u2, zi, z2,t) is some vector function to be defined. Substituting this expression into (1)-(2), we obtain

dh2(u1,u2 ,zi,z2,t) dt

= h.2(ui,u2,zi,z2,t)

Q + Pi (expjUi G*(zi) - 1) Si (t)A-

-MYÏ Pi (ui + zia1i) Si (t)I

(7)

h2(ui,u2,zi,z2,to) = r. Let us use the following substitutions

-1 = e2, bi = qib, te2 = t, Si(t) =Si (t), ui = exh zi = eyi, qib

h2(ui,u2,zi,z2,t) = f2(xi,x2,yi ,y2,T,e),

TO

where bi = f(1 — Bi(x))dx, i = 1,2. Using these notations, problem (7) can be rewritten in the

form

2 Î2(xi,X2,yi,V2 ,T,e)

£ ---= h(xi,X2,yi,y2,T,e)

dT

Q + J2Pi jG(zi) - 1) Si (t)A-

i=i

Pi(ui + zi an) Si (t)I

(8)

Let us find the asymptotic solution (where e ^ 0) of the problem. Let e ^ 0. Then we obtain

f2{xi,x2,yi,y2, t)Q = 0.

Comparing this equation with the first one in (3), we can conclude that f2(x1,x2,y1,y2, t) can be expressed as

Î2(xi,X2,y\,y2, t) = r^2(xi ,X2,yi,y2, t), where &2(x1,x2,y1,y2, t) is some scalar function that satisfies the condition

$2(xi,x2 ,yi,y2, To ) = 1.

So function f2(x1,x2, y1,y2, t, e) can be represented in the expansion form

2

f2(xi,X2,yi,y2,T,£) = &2(xi,X2,yi,y2, t)

' + 53Pi (j£Xi + j£yiaii) Si (t)d

+ O(e2). (9)

Then substituting this expansion into equation (8), we obtain

o(e2) = ®2(xi,x2,yi,y2, t)

rQ + 53Pi (j£Xi + jeyiaii) r (A - kiI) Si (t)+

i=i

2 ] + 53Pi (j£Xi + j£yiaii) Si (t)dQ ,

where d is some row vector that satisfies the following system

{ dQ + r (A — KiI) = 0, de = 0.

We multiply equation (8) by vector e and assume that e ^ 0. The solution of the latter equation with the available initial condition §2(x1, x2, y1, y2, t) = 1 gives

&2(xi,X2,yi,y2,T) = exp^ K1 Pi(xi2 + a2iyi2 + 2xiyiau2) J Si (z)dz+

+j2k2

j K11

I i=1

2 CT r^2 2 2 rT~

Y^Pi'ixi + yiaii)2 Si (z)dz ^PiPg(xi + yiaii)(xg + ygaig) Si (z)Sg(z)dz

„- — 1 Jrn J T0

i=i JTa i=i g=i

g=i

Performing inverse substitutions, we obtain the following expression for the asymptotic characteristic function of the process {k(t), mi(t), m2(t), wi(t), w2(t)}:

{ 2 j2 2

h(2)(ui ,U2 ,zi,z2,t) = exp j Pi(ui + ziau)bi + — Ki ^ pi(u2 + z2a2i)b2 i+ 2

+j2Ki XIPiuizialibi + jK2

i=i

2 -2 2

j 2 2 2

—Ki2_^PiU ~ '

2 2 2 2 i=i

y^P,2 (Ui + ziaii)bi + PiPg (Ui + ziaii)(Ug + zg aig )bibg

i=i i=i g=i

g=i

The theorem is proved. □

It is clear from the form of characteristic function that the four-dimensional process is asymptotically Gaussian.

4. Numerical example

Result of Theorem 3.2 is obtained subject to the asymptotic condition bi ^ x> (e ^ 0). Therefore, it may be used just as an approximation when bi is large enough. To test its practical applicability, we considered several numerical examples with various system parameters (including the distributions of the service time and of the customer capacity). Since all simulation sets led to similar results, for sake of brevity, we discuss in detail just one of them. In particular, we assume that the input MMPP is characterized by the following matrices

-11 5 6 1 0 0

Q = 0.5 -1 0.5 , k = 0 1.5 0

2.5 2.5 -5 0 0 2

The type of incoming customer is defined as i-type with probabilities p1 = 0.4, p2 = 0.6. Volumes of customers has exponential distribution with the rate equal to 2 for type 1 and with the rate equal to 1 for type 2. Service time has gamma distribution with shape and inverse scale parameters a1 = 1.5 and a2 = 0.5, b1 = a1/N, b2 = a2/N, N = 1,10,15, 20, 50,100.

Our goal is to find a lower bound of parameter N to obtain applicable approximation. To find it, we carried out series of simulations for increasing values of N and compared the asymptotic distributions with empiric ones by using the Kolmogorov distance

A = sup\F (x) - A (x)\

x

as an accuracy measure. Here F(x) is the cumulative distribution function build on the basis of simulation results, and A(x) is the Gaussian approximation based on Theorem 2. Values of the Kolmogorov distance for the total capacity of different types of customers for various values of parameter N are presented in Tab. 1.

Table 1. Kolmogorov distances Ai, A2 between simulation results and asymptotic values for the total capacity of customers of type 1 and 2

N 1 10 15 20 50 100

Ai 0.243 0.091 0.032 0.021 0.017 0.012

A2 0.363 0.034 0.025 0.019 0.013 0.010

One can notice that asymptotic results become more accurate when parameter N increases. If we suppose that an approximation is applicable when its Kolmogorov distance is less than 0.03, then we can conclude that asymptotic results are applicable when N is about 15 or more (marked by boldface in the tables).

Conclusion

The queue with MMPP arrivals, infinite number of servers and non-exponential service time is considered in the paper. Moreover, random capacities of customers independent of their service time are taken into account. The analysis is performed subject to the asymptotic condition of an infinitely growing service time. It was shown that multidimensional probability distribution of the number of customers and total capacity in the system is four-dimensional Gaussian distribution subject to this asymptotic condition. Numerical results show that asymptotic results are sufficiently accurate when scale parameter of service time N is greater than 15.

References

[1] C.A.Knessl, On the Sojourn Time Distribution in a Finite Capacity Processor Shared Queue, J. ACM, 40(1993), no. 5, 1238-1301.

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[3] A.Moiseev, A.Demin, V.Dorofeev, V.Sorokin, Discrete-event approach to simulation of queueing networks, Key Engineering Materials, (2016), no. 685, 939-942.

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[6] A.Moiseev, ANazarov, Queueing Network MAP — (GI — x)K with High-rate Arrivals, European Journal of Operational Research, 254(2016), 161-168.

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Ресурсные неоднородные СМО MMPP/GI(2)/ro

Екатерина В. Панкратова

Институт проблем управления РАН Профсоюзная, 65, Москва, 117997

Россия

Светлана П. Моисеева

Томский государственный университет Ленина, 36, Томск, 634050 Россия

Маис П. Фархадов

Институт проблем управления РАН Профсоюзная, 65, Москва, 117997

Россия

Александр Н. Моисеев

Томский государственный университет Ленина, 36, Томск, 634050 Россия

Рассматривается неоднородная бесконечнолинейная система массового обслуживания. На вход системы поступает ММРР-поток требований случайного объема, время обслуживания требований не зависит от их объёма. Требования, поступившие в систему, с вероятностью pi (i = 1, 2) определяются как требования типа i. Показано, что совместное распределение вероятностей числа занятых приборов и суммарного объема занятого ресурса в системе является многомерным гауссовским при асимптотическом условии эквивалентного роста времени обслуживания на приборах 'разного типа.Результаты моделирования позволяют определить область применимости асимптотического метода.

Ключевые слова: бесконечнолинейные СМО, случайный объём требований, марковский модулированный пуассоновский процесс.