Bushkova, T. V.

Moiseeva, Svetlana P.

English

Tomsk State University Repository

МИНИСТЕРСТВО НАУКИ И ВЫСШЕГО ОБРАЗОВАНИЯ  РОССИЙСКОЙ ФЕДЕРАЦИИ ТОМСКИЙ ГОСУДАРСТВЕННЫЙ УНИВЕРСИТЕТ МАТЕРИАЛЫ VI Международной молодежной научной конференции «МАТЕМАТИЧЕСКОЕ И ПРОГРАММНОЕ ОБЕСПЕЧЕНИЕ ИНФОРМАЦИОННЫХ, ТЕХНИЧЕСКИХ И ЭКОНОМИЧЕСКИХ СИСТЕМ» Томск, 24–26 мая 2018 г. Под общей редакцией кандидата технических наук И.С. Шмырина Томск Издательский Дом Томского государственного университета 2018   347 СЕКЦИЯ IX. COMPUTER SCIENCE STUDY OF THE MMPP/GI/1 QUEUEING SYSTEM WITH RANDOM CUSTOMERS’ CAPACITIES T.V. Bushkova, S.P. Moiseeva Tomsk State University bushkova70@mail.ru, smoiseeva@mail.ru Introduction Queuing systems represent a powerful mathematical tool for investigating the perfor-mance of a wide variety of real-life systems, ranging from telecommunication networks to financial markets, from computer architectures to supply chain management and airplane traf-fic control, just to cite a few. Analytical tractability of the corresponding models strongly de-pends on the nature of the underlying processes (Poisson arrivals have many nice features that strongly simplify the analysis) and on the system geometry. Although physical resources are always finite, quite often it is easier to study queuing systems in which the corresponding parameters assume infinite values. For instance, the over-flow probability is often used as an upper bound for the loss probability in finite-buffer queues and, indeed, asymptotic results are available even for strongly non-Markovian sys-tems [1]. Moreover, infinite-server queuing systems may be applicable in case ofmodels with a limited number of server devices as described in [2]. In this work, an infinite-server queuing system, fed by non-Poisson arrivals with random customers’ capacities, is considered. 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 the independent quantities (see [3,4] and refer-ences therein). For instance, in [3], performance analysis of LTE (Long Term Evolution) net-works is carried out in terms of flow-level dynamics and the amount of required radio re-sources does not depend on the duration of the flow. Such queues are also important in mod-eling devices, where it is necessary to calculate a sufficient volume of buffer for data stor-ing [5,6]. The results for single-server queues with limited buffer and LIFO (last in, first out) service discipline were presented in [7], where algorithms for the calculation of stationary characteristics were derived. A new trend in the study of queuing systems is the analysis of the systems with non-Poisson arrivals and onexponential service time. So, in the works [2,8–11], queues with re-newal arrivals, Markovian Arrival processes (MAP), and MMPP are studied under various asymptotic conditions. The main contribution of this paper consists in extending such analy-sis, focusing on the properties of the bidimensional process describing the number of custom-ers and the total capacity in the system when an infinite-server queue is fed by MMPP arrivals with randomcapacities and nonexponential service time distribution. 1. Matematical model Consider a queue with infinite number of servers (Fig. 1) and assume that customers ar-rive according to an MMPP.  348  Fig. 1. Queue MMPP/GI/∞ with random customers capacities The input process is defined by its generator matrix ijQ q=  of size K×K and the condi-tional rates 1,..., Кλ λ , typically composed into the diagonal matrix { }1diag ,...., КΛ = λ λ . De-note the underlying Markov chain of the MMPP as ( ) 1,2,...,k t K∈ . Let each customer has some random capacity 0v >  with distribution function ( )G y . An arriving customer instantly occupies a server in the system and its service time has distribution function ( )B x ; when the service is completed, the customer leaves the system. Customers’ capacities and service times are mutually independent and do not dependent on the epochs of customers’ arrivals. Denote by ( )i t  and ( )V t  the number of customers in the system at time t and their total capacity, respectively. Let us obtain the probabilistic characteristics of two-dimensional process ( ) ( ){ };i t V t . This process is not Markovian; therefore, the dynamic screening method has been used for its investigation. Consider two time axes that are numbered as 0 and 1 (Fig. 2). Let axis 0 shows the epochs of customers’ arrivals, while axis 1 corresponds to the screened process. Let us introduce a function ( )S t  (dynamic probability) that satisfies the condition ( )0 1S t≤ ≤ . Let us assume that a customer, arriving in the system at time t, is screened to axis 1 with probability ( )S t , and not screened with probability ( )1 S t− . Let the system be empty at moment t0 and let us fix some arbitrary moment T in the fu-ture. ( )S t  represents the probability that a customer arriving at time t will be serviced in the system by the moment T. It is easy to show [11] that ( ) ( )1S t B T t= − −  for 0t t T≤ ≤ . Denote by ( )n t  and ( )W t  the number of arrivals screened before the moment t on axis 1 and their total capacity, respectively.  Fig. 2. Screening of the customers’ arrivals As it is shown in [9], the probability distribution of the number of customers in the sys-tem at the moment T coincides with the probability distribution of the number of screened arrivals on the axis:  ( ){ } ( ){ }P i T m P n T m= = =  for all 1,2,...m = . It is easy to prove the same property for the extended process ( ) ( ){ },i t V t :  349  ( ){ } ( ){ }, ( ) , ( )P i T m V T z P n T m W T z= < = = <  (1) for all 1,2,...m =  and 0z ≥ . Let us use Eqs. (1) for the investigation of the process ( ) ( ){ },i t V t  via the analysis of the process ( ) ( ){ },n t W t . 2. Kolmogorov differential equations Let us consider the three-dimensional Markovian process ( ) ( ) ( ){ }, ,k t n t W t . Denoting the probability distribution of this process by { }( , , , ) ( ) , ( ) , ( ) )P k n w t P k t k n t n W t w= = = 〈  and taking into account the formula of total probability, one can write the following system of Kolmogorov differential equations:  ( ) ( ) ( ) ( ) ( ) ( )0, , , , 1, , , , , , , ,zk vkvP k n w t S t P k n w y t dG y P k n w t P v n w t qt  ∂ = λ − − − + ∂   ∑∫  for 1,k K= ; 1,2,...n = ; 0w > . Let us introduce the partial characteristic function:  ( ) ( )1 21 2 0 0, , , , , ,ju n ju wnh k u u t e e P k n dw t∞∞==∑ ∫  where 1j = −  is the imaginary unit. Then, one can write the following equations:  ( ) ( ) ( ) ( )( ) ( )11 2 *1 2 2, , , , , , 1 , , ,juk vkvh k u u t h k u u t S t e G u h k n w t qt∂ = λ − +∂ ∑  for 1,k K=  where ( ) ( )* 0 juyG u e dG y∞= ∫ . Let us rewrite this system in the matrix form:  ( ) ( ) ( ) ( )( )11 2 *1 2 2, , , , 1juh u u t h u u t S t e G u Qt∂  = Λ − + ∂  (2) with the initial condition  ( )1 2 0, ,h u u t r=  (3) where ( ) ( ) ( )1 2 1 2 1 2, , 1, , , ,...., , , ,h u u t h u u t h K u u t=     and ( ) ( )1 ,....,r r r K=     represents the stationary distribution of the underlying Markov chain, i.e., vector r satisfies the following linear system:  0,1,rQre = =  (4) where e is the column vector with all entries equal to 1. 3. Asymptotic analysis In general, the exact solution of Eq. (2) is not available, but it may be found under asymptotic conditions. In this paper, the case of infinitely growing service time is considered. Denote by  ( ) ( )( )1 0 0 1b xdB x B x dx∞ ∞= = −∫ ∫  the mean service time; then, the asymptotic condition is 1b →∞ . Let us solve Problem (2)–(3) under such asymptotic condition and we obtain approx-imate solutions with different order of accuracy, named as “first-order asymptotic” ( ) ( )(1)1 2 1 2, , , ,h u u t h u u t≈  and “second-order asymptotic” ( ) ( )(2)1 2 1 2, , , ,h u u t h u u t≈ .  350 3.1. First-order asymptotic analysis Let us formulate and prove the following statement. Lemma. The first-order asymptotic characteristic function of the probability distribution of the process { }( ), ( ), ( )k t n t W t  has the form:  ( ) ( ) ( )0(1) 1 2 1 1 2 1 1, , exp tth u u t r ju k ju k a S v dv  = +   ∫  where 1k r e= Λ  and ( )1 0a ydG y∞= ∫ ) is the mean customer capacity. Proof. By performing the substitutions  11bε = , tε = τ , 0 0tε = τ , 1 1u x= ε , 2 2u x= ε , ( ) ( )1S t S= τ , ( ) ( )1 2 1 1 2, , , , ,h u u t f x x= τ ε  in expressions (2) and (3), one obtains  ( ) ( ) ( ) ( )( )11 1 2 *1 1 2 1 2, , , , , , 1j xf x x f x x S e G x Qε∂ τ ε  ε = τ ε Λ τ ε − + ∂τ  (5) with the initial condition  ( )1 1 2 0, , ,f x x rτ ε = . (6) Let us find the asymptotic solution of Problem (5)–(6) ( ) ( )1 1 2 1 1 20, , lim , , ,f x x f x xε→τ = τ ε  in two steps. Step 1. Let 0ε→  in (5)–(6); then, one obtains the following system of equations:  ( )( )1 1 21 1 2 0, , 0;, , .f x x Qf x x Q rτ = τ =  Taking into account (4), one can conclude that ( )1 1 2, ,f x x τ  can be expressed as  ( ) ( )1 1 2 1 1 2, , , ,f x x r x xτ = Φ τ  (7) where ( )1 1 2, ,x xΦ τ  is some scalar function which satisfies the condition  1 1 2 0( , , ) 1x xΦ τ = . (8) Step 2. Let us multiply (5) by vector e, substitute (7), divide the result by ε , and per-formthe asymptotic transition 0ε→ . Then, taking into account that 0Qe =  and 1re = , one obtains the following differential equation for the function ( )1 1 2, ,x xΦ τ :  ( ) ( ) ( ) ( )1 1 2 1 1 2 1 1 1 1 1 1, , , ,x x x x S jx k jx k a∂Φ τ = Φ τ τ +∂τ . (9) The solution of Problem (8)–(9) is as follows:  ( ) ( ) ( )01 1 2 1 1 21 1 1 1, , expx x jx k jx k a S v dvττ  Φ τ = +   ∫ . Substituting this expression into (7), one obtains  ( ) ( ) ( )01 1 2 1 1 2 1 1 1, , expf x x r jx k jx k a S v dvττ  τ = +   ∫ . Therefore, one can write  ( ) ( ) ( ) ( ) ( )01 1 2 1 1 2 1 1 2 1 1 2 1 1 1, , , , , , , exph u u t f x x f x x r jx k jx k a S v dvττ  = τ ε ≈ τ = + =   ∫   351  ( ) ( )01 1 2 1 1 1expr ju k ju k a S v dvττ  = +   ∫ . Thus, the proof is complete. 3.2. Second-order asymptotic analysis The main result is the following theorem. Theorem. The second-order asymptotic characteristic function of the probability distri-bution of the process { }( ), ( ), ( )k t n t W t  has the form:  ( ) ( ) ( ) ( ) ( ) ( )0 0 022 1 21 2 1 1 2 1 1 1 1 1, , exp 2t t tt t tjuh u u t r ju k ju k a S v dv k S v dv k S v dv    = + + + +        ∫ ∫ ∫   ( ) ( ) ( ) ( ) ( )0 0 0 022 2 2 21 2 2 1 1 2 1 1 2 12 t t t tt t t tju k a S v dv k a S v dv ju ju k a S v dv k a S v dv   + + + +         ∫ ∫ ∫ ∫ , where ( )2 12k g k I e= Λ − , ( )22 0a y dG y∞= ∫  and the row vector g satisfies the linear matrix system  ( )1 ,const .gQ r k Ige = − Λ =  Proof. Let ( )2 1 2,h x x t  be a vector function that satisfies the equation:  ( ) ( ) ( ) 01 2 2 1 2 1 1 2 1 1, , , , exp ( )tth u u t h u u t ju k ju k a S v dv  = +   ∫ . (10) Substituting this expression into (2) and (3), one obtains the following problem:  ( ) ( ) ( )( ) ( ) ( ) ( )12 1 2 *2 1 2 2 1 1 2 1 1, , , , 1juh u u t h u u t e G u S t ju k ju k a S t I Qt∂  = − Λ − + + ∂  (11) with the initial condition  ( )2 1 2 0, ,h u u t r=  (12) where I is the identity matrix. Let us make the substitutions:  ( ) ( ) ( ) ( )2 2 2 0 0 1 1 1 11 1 2 1 2 2 1 21 ; ; ; ; ;; , , , , , .t t u x u xbS t S t h u u t f x xε = ε = τ ε = τ = ε = ε= = τ ε  (13) Using these notations, Problem (11)–(12) can be rewritten in the form  ( ) ( ) ( ) ( )( ) (12 1 22 *2 1 2 1 2 1 1, , , , , , 1j xf x x f x x S e G x j k xε∂ τ ε ε = τ ε Λ τ ε − − ε +∂τ   ) ( )1 2 1 1j k x a S I Q+ ε τ +   (14) with the initial condition  ( )2 1 2 0, , ,f x x rτ ε = . (15) Let us find the asymptotic solution of this problem ( ) ( )2 1 2 2 1 20, , lim , , ,f x x f x xε→τ = τ ε  in three steps. Step 1. Letting 0ε→  in (14)–(15), one obtains the following system of equations:  352  ( )( )2 1 22 1 2 0, , 0;, , .f x x Qf x x r τ = τ =  Therefore, taking into account (4), one can write:  ( ) ( )2 1 2 2 1 2, , , ,f x x r x xτ = Φ τ  (16) where ( )2 1 2, ,x xΦ τ  is some scalar function which satisfies the condition  ( )2 1 2 0, , 1x xΦ τ = . (17) Step 2. Using (16), the function ( )2 1 2, ,f x x τ  can be represented in the expansion form:  ( ) ( ) ( )( ) ( )22 1 2 2 1 2 1 1 2 1, , , , ,f x x x x r gS j x j x a Oτ ε = Φ τ + τ ε + ε + ε    (18) where g is the row vector that satisfies the condition constge =  and ( )2O ε  is the row vector of the secondorder infinitesimals. Let us use substitution (18) and the expansion ( )21j xe j x Oε = + ε + ε . In Eq. (14). Taking into account (4) and making the transition 0ε→ , one obtains the fol-lowing matrix equation for the vector g: ( )1gQ r k I= − Λ . Step 3. Let us multiply Eq. (14) by vector e and use expression (18) and the second-order expansion: ( ) ( )2 31 2j x j xe j x Oε ε= + ε + + ε . After some transformations, using the notation ( )2 12k g k I e= Λ − , one obtains the fol-lowing differential equation for the function ( )2 1 2, ,x xΦ τ :  ( ) ( ) ( ) ( ) ( )( ) ( ) ( )(2 22 1 2 1 222 1 2 1 1 2 1 1 2 1, , , , 2 2x x jx jxx x k S k S k a S∂Φ τ = Φ τ τ + τ + τ +∂τ    ( )) ( ) ( )( )2 2 22 1 1 1 2 1 1 1 2 1 1 .k a S jx jx k a S k a S + τ + τ + τ   The solution of this equation with initial condition (17) is as follows:  ( ) ( ) ( ) ( ) ( ) ( )10 0 02 21 2 22 1 2 1 1 2 1 2 1, , exp 2 2jx jxx x k S v dv k S v dv k a S v dvτ τ ττ τ τ   Φ τ = + + +         ∫ ∫ ∫   ( ) ( ) ( )1 10 0 02 2 22 1 1 2 1 1 1 2 1k a S v dv jx jx k a S v dv k a S v dvτ τ ττ τ τ   + + +      ∫ ∫ ∫ . Substituting this expression in formula (16) and performing the substitutions that are in-verse to (13) and (10), one obtains  ( ) ( ) ( ) ( ) ( ) ( )0 0 022 1 21 2 1 1 2 1 1 1 2, , exp 2t t tt t tjuh u u t r ju k ju k a S v dv k S v dv k S v dv  = + + + +      ∫ ∫ ∫  ( ) ( ) ( ) ( ) ( )0 0 0 022 2 2 21 2 2 1 1 2 1 1 2 12 t t t tt t t tju k a S v dv k a S v dv ju ju k a S v dv k a S v dv    + + + +         ∫ ∫ ∫ ∫  for the asymptotic characteristic function of the process ( ) ( ) ( ){ }, ,k t n t W t . The proof is complete. Corollary. Assuming t T=  and 0t →∞  and using Eqs. (1), one obtains the steady-state characteristic function of the process under study ( ) ( ){ },i t V t :  353  ( ) ( ) ( ) ( ) ( ) ( )( )}2 21 2 21 2 1 1 1 2 1 1 1 1 1 2 2 1 2 1 2 1 21 2 1 1 1 2 1 2, exp 2 2.ju juh u u ju k b ju k a b k b k b k a b k a bju ju k a b k a b= + + + + + + + +  (19) where ( )( )1 0 1b B v dv∞= −∫ , ( )( )22 0 1b B v dv∞= −∫ . From the form of the characteristic function (19), it is clear that the probability distribu-tion of the two dimensional process ( ) ( ){ },i t V t  is asymptotically Gaussian with vector of means [ ]1 1 1 1 1,a k b k a b=  and covariance matrix  1 22 12 1 1 2 2 1 1 1 2 1 22 212 1 1 1 2 1 2 1 2 1 2 1 2K k b k b k a b k a bK K k a b k a b k a b k a b σ + + = =   σ + +   . Therefore, the correlation coefficient is given by  12 1 1 1 2 1 2 21 2 1 1 2 2 1 2 1 2 1 2K k a b k a br k b k b k a b k a b+= =σ σ + + . Conclusion In the paper, the queue with MMPP arrivals, infinite number of servers, and nonexponen-tial service time is considered. Moreover, random customers’ capacities, independent of their service time, are assumed. The analysis is performed under the asymptotic condition of an infinitely growing service time. 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Study of the MMPP/GI/1 queueing system with random customers' capacities

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Study of the MMPP/GI/1 queueing system with random customers' capacities

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