In the paper, the infinite-server queueing system with a random capacity of customers is considered. In this system, the total capacity of customers is analysed by means of the asymptotic analysis method with high-rate Markov Modulated Poisson Process arrivals. It is obtained that the stationary probability distribution of the total customer capacity can be approximated by the Gaussian distribution. Parameters of the approximation is also derived in the pape

Lisovskaya, Ekaterina

Moiseeva, Svetlana

Pagano, Michele

English

Tomsk State University Repository

UDC 519.872The total capacity of customers in the MMPP/GI/∞queueing systemE. Lisovskaya∗, S. Moiseeva∗, M. Pagano†∗Department of probability theory and mathematical statistics,Tomsk State UniversityLenin ave. 36, Tomsk, 634050, Russia†Department of Internation Engeneering,University of PisaVia Caruso 16, Pisa, 56122, ItalyAbstract. In the paper, the infinite-server queueing system with a random ca-pacity of customers is considered. In this system, the total capacity of customersis analysed by means of the asymptotic analysis method with high-rate MarkovModulated Poisson Process arrivals. It is obtained that the stationary proba-bility distribution of the total customer capacity can be approximated by theGaussian distribution. Parameters of the approximation is also derived in thepaper.Keywords: infinite-server queueing system, customer with random capacity,Markovian Modulated Poisson Process.1. IntroductionIn the design of messages processing and transmission systems, deter-mining the memory capacity required for information storage is a relevantopen issue [1,2]. The total capacity is a random quantity and in queueingtheory it is given by the sum of the lengths of all messages, which arewaiting in the buffer or currently processed by servers.Since in real systems customers are heterogeneous (for instance in com-puter networks packet size may vary from a few tens of bytes to 1500 bytesin case of Ethernet links), this paper focuses on the analysis of queueingsystem (QS) with random customer capacity. The main classes of modelsused for such models and their applications in real information systemsare given [1, 2].There are several works on the study of such systems with Poissonarrival process and service time independent of the customer capacity. Forexample, in [3] for systems with limited total capacity the generalization ofthe Erlang problem is considered in stationary conditions. Moreover, in [4,5], the stationary distribution of the customers number and the probabilityof losses are obtained for systems with limited memory. Finally, in [9], theauthors consider systems with service time depended on the customerscapacity or with waiting time restrictions, an assumption very relevant forreal-time applications.It is important to point out that most known results are obtained forqueueing systems and networks with Poisson arrivals. Unfortunately, itLisovskaya E., Moiseeva S., PaganoM. 313has been proved that the Poisson model is suitable only for few casesof modern telecommunication streams [6] and, in general, the correlationamong arrivals must be taken into account. Therefore, many researchesuse more complex models of arrivals, such as Markovian Arrival Processes(MAPs) [7] or semi-Markov processes [8].The main contribution of this paper consists in extending previousworks on random capacity customers to the case of correlated arrivals. Inmore detail, the problem statement is formally defined in section 2 and insection 3 the corresponding Kolmogorov differential equations are derived.Then section 4 presents the results of the asymptotic analysis, focusingon first and second order approximations. Finally, the main findings aresummarized in section 5.2. Problem statementIn this paper, the MMPP/GI/∞ QS with random capacity customersis studied. The arrival process is a Markov Modulated Poisson Process(MMPP), a widely-used special case of MAP [7]. The system has anunlimited number of servers and service times on each server are i.i.d. withdistribution function B (x). All customers have a random capacity ν > 0with the probability distribution G (y) = P {ν < y} and the customerscapacity are independent. Moreover, we assume that service time andcustomers capacity are mutually independent. After the service, customersleave the system and carry out the capacity.Let us denote the number of customers in the system and the totalcustomers capacity at time t by i (t) and V (t), respectively. We considertwo-dimensional stochastic process {i (t) , V (t)}, which is not Markovian.Therefore, we propose the dynamic screening method for its investigation[10].For the screened process construction, we fix some point in time T . Weassume that the customer arrived in the system at time t < T creates apoint in the screened process with probabilityS (t) = 1−B (T − t)or does not create it with probability 1 − S (t). We name the pointsoccurred in the screened process before t as customers in the screenedprocess at time t.Let us denote the customers number in the screened process at themoment t by n (t). Then, if at the initial moment t0 < T the system isempty, we have the following equality at the moment T :P {i (T ) = m} = P {n (T ) = m} .Note that this method exactly determines the characteristics of theprocess V (t) since the screened process contains only customers which donot finish the service at the moment T .314 DCCN—20163. The system of Kolmogorov differential equationsLet us consider the three-dimensional Markovian process{k (t) , n (t) , V (t)}, where k (t) identifies the state of the modulat-ing Markov chain of the MMPP input process at time t (1 ≤ k (t) ≤ K),which is defined through the infinitesimal generator matrix Q and ratematrix Λ:Q =q11 q12 . . . q1Kq21 q22 . . . q2K. . . . . . . . . . . .qK1 qK2 . . . qKK , Λ =λ1 0 . . . 00 λ2 . . . 0. . . . . . . . . . . .0 0 . . . λK .Denoting the probability distribution of this process by P (k, n, z, t) =P {k (t) = k, n (t) = n, V (t) < z}, we can write the corresponding systemof Kolmogorov differential equations taking into account the formula oftotal probability:P (k, n, z, t+ ∆t) = P (k, n, z, t) (1− λk) (1− qkk) +P (k, n, z, t)λk∆t (1− S (t)) +λk∆tS (t)z∫0P (k, n− 1, z − y, t) dG (y) + (1)∑ν 6=kqνk∆tP (ν, n, z, t) + o (∆t) ,for k = 1 . . .K, n = 0, 1, 2, . . . , z > 0.From (1), we obtain the system of Kolmogorov differential equations∂P (k, n, z, t)∂t= λkS (t) z∫0P (k, n− 1, z − y, t) dG (y)+∑νqνkP (ν, n, z, t)for k = 1 . . .K, n = 0, 1, 2, . . . , z > 0.We introduce a partial characteristic function of the form:H (k, u1, u2, t) = M {exp (ju1n (t) + ju2V (t))} =∞∑n=0eju1n∞∫0eju2zP (k, n, z, t) dzLisovskaya E., Moiseeva S., PaganoM. 315for k = 1 . . .K, n = 0, 1, 2, . . . , z > 0.Considering that∞∑n=0eju1n∞∫0eju2zz∫0P (k, n− 1, z − y, t) dG (y) dz =eju1H (k, u1, u2, t)G∗ (u2) ,whereG∗ (u2) =∞∫0eju2ydG (y) , (2)we can write the following system of equations:∂H (k, u1, u2, t)∂t= λkS (t)H (k, u1, u2, t)[eju1G∗ (u2)− 1]for k = 1 . . .K.We write this system in the form of a matrix equation∂H (u1, u2, t)∂t= H (u1, u2, t)[ΛS (t)(eju1G∗ (u2)− 1)+ Q](3)with the initial conditionH (u1, u2, t0) = r, (4)whereH (u1, u2, t) = [H (1, u1, u2, t) , H (2, u1, u2, t) , . . . ,H (K,u1, u2, t)] ,andr = [r (1) , r (2) , . . . , r (K)] ,is the row vector of the stationary distribution of the modulating Markovchain: {rQ = 0,re = 1,(5)e being a column vector with all entries equal to 1.316 DCCN—20164. The asymptotic analysis methodThe exact solution of the equation (3) is, in general, not available, butit is possible to get asymptotic results in case of heavy loads. To thisaim we will use the asymptotic analysis method under the condition of aninfinitely growing arrival rate. Let us substitute Λ = NΛ1 and Q = NQ1into the equation (3), where N is some parameter which will be used forthe asymptotic analysis (N →∞ in theoretical studies).Then, the equation (3) takes the form1N∂H (u1, u2, t)∂t= H (u1, u2, t)[Λ1S (t)(eju1G∗ (u2)− 1)+ Q1](6)with the initial conditionH (u1, u2, t0) = r. (7)4.1. The first-order asymptotic analysisThe main result is summarized by the following lemma.Lemma. The first-order asymptotic characteristic function of theprobability distribution of the process {k (t) , n (t) , V (t)} has the formH (u1, u2, t) = r expNλ [ju1 + ju2a1]t∫t0S (τ) dτ ,where the row vector r is defined by the system of linear equations (5), λdenotes the average rateλ = rΛeand a1 is the mean of the random variable defining the customer capacitya1 =∞∫0ydG (y) .Proof.Let us perform the substitutionsε =1N, u1 = εw1, u2 = εw2,H (u1, u2, t) = F1 (w1, w2, t, ε) (8)in the expressions (5) and (6).Then the problem (5) – (6) takes the formLisovskaya E., Moiseeva S., PaganoM. 317ε∂F1 (w1, w2, t, ε)∂t=F1 (w1, w2, t, ε)[Λ1S (t)(ejεw1G∗ (εw2)− 1)+ Q1](9)with the initial conditionF1 (w1, w2, t, ε) = r.Let us find the asymptotic solution (where ε → 0) of the problem(8) – (9), i.e. the F1 (w1, w2, t) = limε→0F1 (w1, w2, t, ε).Step 1. Letting ε→ 0 in (9), we obtainF1 (w1, w2, t) Q1 = 0.Comparing this equation with the first one in (5), we can conclude thatF1 (w1, w2, t) can be expressed asF1 (w1, w2, t) = rΦ1 (w1, w2, t) , (10)where Φ1 (w1, w2, t) is some scalar function which satisfies the conditionΦ1 (w1, w2, t0) = 1.Step 2. Let us multiply (9) by vector e, substitute (10), divide theresults by ε and perform the asymptotic transition ε → 0. Then, takinginto account that Q1e = 0 and re = 1, we obtain the following differentialequation for the function Φ1 (w1, w2, t)∂Φ1 (w1, w2, t)∂t= Φ1 (w1, w2, t) [λS (t) (jw1 + jw2a1)] . (11)The solution of (11) with the initial condition givesΦ1 (w1, w2, t) = expλ (jw1 + jw2a1)t∫t0S (τ) dτ .Substituting this expression into (10), we obtainF1 (w1, w2, t) = r expλ (jw1 + jw2a1)t∫t0S (τ) dτ .318 DCCN—2016Using substitutions (8), we can write the asymptotic (as ε→ 0) equal-ity:H (u1, u2, t) = F1 (w1, w2, t, ε) ≈ F1 (w1, w2, t) = rΦ1 (w1, w2, t) == r expλ [j u1ε + j u2ε a1]t∫t0S (τ) dτ =r expNλ [ju1 + ju2a1]t∫t0S (τ) dτ .The proof is complete.Corollary. When t = T we obtain the characteristic function of theprocess {i (t) , V (t)} in the steady state regimeH (u1, u2, t) = exp {Nλb1 [ju1 + ju2a1]} ,whereb1 =T∫−∞S (τ) dτ =T∫−∞(1−B (T − τ)) dτ =∞∫0(1−B (τ)) dτdenotes the mean service time.4.2. The second-order asymptotic analysisThe main result is summarized by the following theorem.Theorem. The second-order asymptotic characteristic function of theprobability distribution of the process {k (t) , n (t) , V (t)} has the formH (u1, u2, t) = r expNλ (ju1 + ju2a1)t∫t0S (τ) dτ+(ju1)22Nλ t∫t0S (τ) dτ +Nκt∫t0S2 (τ) dτ+(ju2)22Nλa2 t∫t0S (τ) dτ +Nκa21t∫t0S2 (τ) dτ+Lisovskaya E., Moiseeva S., PaganoM. 319j2u1u2Nλa1 t∫t0S (τ) dτ +Nκa1t∫t0S2 (τ) dτ ,whereκ = 2g(Λ1 − λI) e,and the row vector g satisfies the linear matrix systemgQ1 = r(λI−Λ1) ,ge = 1.Proof.Denote by H2 (u1, u2, t) a multi-dimensional function that satisfies theequationH (u1, u2, t) = H2 (u1, u2, t) expNλ (ju1 + ju2a1)t∫t0S (τ) dτ . (12)Substituting this expression into (6) and (7), we obtain the followingproblem:1N∂H2 (u1, u2, t)∂t+ λ (ju1 + ju2a1)S (t) H2 (u1, u2, t) = (13)= H2 (u1, u2, t)[Λ1S (t)(eju1G∗ (u2)− 1)+ Q1],with the initial conditionH2 (u1, u2, t0) = r. (14)Let us perform the following substitutionsε2 =1N, u1 = εw1, u2 = εw2,H2 (u1, u2, t) = F2 (w1, w2, t, ε) . (15)Using these notations the problem (13) – (14) can be rewritten in the formε2∂F2 (w1, w2, t, ε)∂t+ F2 (w1, w2, t, ε)λ (jεw1 + jεw2a1)S (t) = (16)= F2 (w1, w2, t, ε)[Λ1S (t)(ejεw1G∗ (εw2)− 1)+ Q1],with the initial conditionF2 (w1, w2, t0, ε) = r. (17)320 DCCN—2016Let us find the asymptotic solution (as ε→ 0) of this problem, i.e. thefunction F2 (w1, w2, t) = limε→0F2 (w1, w2, t, ε) .Step 1. Letting ε→ 0 in (16) – (17), we obtain the following system ofequations: {F2 (w1, w2, t) Q1 = 0,F2 (w1, w2, t0) = r,Then, using (4), we can writeF2 (w1, w2, t) = rΦ2 (w1, w2, t) , (18)where Φ2 (w1, w2, t) is some scalar function which satisfies the conditionΦ2 (w1, w2, t0) = 1.Step 2. Using (18), the function F2 (w1, w2, t) can be represented inthe expansion formF2 (w1, w2, t, ε) =Φ2 (w1, w2, t) [r + g (jεw1 + jεw2a1)S (t)] + O(ε2), (19)where g is some row vector which satisfying the condition ge = 1 andO(ε2)is row vector whose elements are infinitesimals of the same orderas ε2.Let us use the substitution (19) and the Taylor-Maclaurin expansionsejεw1 = 1 + jεw1 +O(ε2), ejεw2 = 1 + jεw2 +O(ε2)in (16). Considering the (2), we perform in the obtained equality of thelimiting transition ε→ 0, we obtain matrix equation for the vector ggQ1 = r(λI−Λ1) ,where I is diagonal unit matrix.Step 3. We multiply the (16) by the e, using (19) and the second-orderexpansionsejεw1 = 1 + jεw1 +(jεw1)22+O(ε2),ejεw2 = 1 + jεw2 +(jεw2)22+O(ε2).As a result of simple transformations with the notationκ = 2g(Λ1 − λI) e,Lisovskaya E., Moiseeva S., PaganoM. 321we obtain the following differential equation for the function Φ2 (w1, w2, t)∂Φ2 (w1, w2, t)∂t= Φ2 (w1, w2, t){(jw1)22(λS (t) + κS2 (t))+(jw2)22(λa2S (t) + κa21S2 (t))+ j2w1w2(λa1S (t) + κa1S2 (t))},where a2 =∞∫0y2dG (y) is the second moment of the random customercapacity ν.The solution of the latter equation with the available initial conditionΦ2 (w1, w2, t0) = 1 gives the expression Φ2 (w1, w2, t)Φ2 (w1, w2, t) = exp (jw1)22λ t∫t0S (τ) dτ + κt∫t0S2 (τ) dτ+(jw2)22λa2 t∫t0S (τ) dτ + κa21t∫t0S2 (τ) dτ+j2w1w2λa1 t∫t0S (τ) dτ + κa1t∫t0S2 (τ) dτ ,and substituting in (18) we obtainF2 (w1, w2, t) = r exp (jw1)22λ t∫t0S (τ) dτ + κt∫t0S2 (τ) dτ+(jw2)22λa2 t∫t0S (τ) dτ + κa21t∫t0S2 (τ) dτ+ (20)j2w1w2λa1 t∫t0S (τ) dτ + κa1t∫t0S2 (τ) dτ .Performing in (20) the substitutions inverse to (15) and (12), we obtainthe following expression for the asymptotic characteristic function of the322 DCCN—2016number of customers of screened process and total capacity of customersat the moment t:H (u1, u2, t) = r expNλ (ju1 + ju2a1)t∫t0S (τ) dτ+(ju1)22Nλ t∫t0S (τ) dτ +Nκt∫t0S2 (τ) dτ+(ju2)22Nλa2 t∫t0S (τ) dτ +Nκa21t∫t0S2 (τ) dτ+j2u1u2Nλa1 t∫t0S (τ) dτ +Nκa1t∫t0S2 (τ) dτ ,whereκ = 2g(Λ1 − λI) e,and the row vector g satisfies the linear matrix systemgQ1 = r(λI−Λ1) ,ge = 1.The proof is complete.Corollary 1. When t = T we obtain the characteristic function of theprocess i (t) , V (t) in the steady state regimeH (u1, u2, t) = exp{Nλ (ju1 + ju2a1) b1 +(ju1)22(Nλb1 +Nκb2) +(ju2)22(Nλa2b1 +Na21κb2)+ j2u1u2 (Nλa1b1 +Nκa1b2)}, (21)whereb2 =T∫−∞S2 (τ) dτ.Lisovskaya E., Moiseeva S., PaganoM. 323From the form of function (21) it is clear that the two-dimensional pro-cess i (t) , V (t) is asymptotically Gaussian with the vector of mathematicalexpectationsa = [Nλb1, Nλa1b1]and the covariance matrixA =[σ21 rσ1σ2rσ1σ2 σ22]=[Nλb1 +Nκb2 Nλa1b1 +Nκa1b2Nλa1b1 +Nκa1b2 Nλa2b1 +Nκa21b2].Corollary 2. The asymptotic characteristic function of the total cus-tomer capacity in the steady-state regime is given by a Gaussian charac-teristic functionH (u, t) = exp{juNλa1b1 +(ju)22(Nλa2b1 +Na21κb2)},with parameters a = Nλa1b1 and σ2 = Nλa2b1 +Na21κb2.5. ConclusionsIn the paper, a queueing system with random customers capacity andservice time independent of its capacity is considered in case of correlatedarrivals, described by an MMPP process. For such system, the total cus-tomers capacity is derived by using the asymptotic analysis method in caseof heavy loads. It is obtained that the stationary probability distributionof total capacity can be approximated by a Gaussian distribution and theparameters of the approximation are derived in the paper.References1. Tikhonenko, O.M. Destricted capacity queueing systems. Determina-tion of their characteristics Autom. Remote Control, 58 (6), 969-972(1997)2. Tikhonenko, O.M. Computer Systems Probability Analysis. Aka-demicka Oficyna Wydawnicza EXIT, Warsaw (2006) (in Polish)3. Romm, E.L., Skitovich, V.V. On certain generalization of problem ofErlang. Avtomat. i Telemekh., No. 6, 164-168 (1971)4. Tikhonenko, O.M. Systems for servicing requests of random length withrestrictions. Avtomat. i Telemekh., No. 10, 126-134 (1991)5. Tikhonenko, O.M. Queuing Systems with Processor Sharing and Lim-ited Resources. Autom. Remote Control, No. 5, 803-815 (2010)6. Heyman, D.P., Lucantoni, D. Modelling multiple IP traffic streamswith rate limits. IEEE/ACM Trans. on Networking, No. 11, 948-958(2003)324 DCCN—20167. Chakravarthy, S.R. Markovian arrival processes. Wiley Encyclopedia ofOperations Research and Management Science (2010)8. Moiseev, A., Nazarov, A. Asymptotic Analysis of the Infinite-ServerQueueing System with High-Rate Semi-Markov Arrivals. IEEE Inter-national Congress on Ultra Modern Telecommunications and ControlSystems (ICUMT 2014), IEEE Press, New York, 507-513 (2014)9. Naumov, V., Samuoylov, K., Sopin, E., Yarkina, N., Andreev, S.,Samuylov, A. LTE Performance Analysis Using Queuing Systems withFinite Resources and Random Requirements. Proc. of 8th Int. Con-gress on Ultra Modern Telecommunications and Control Systems andWorkshops, 100-103 (2015).10. Moiseev A.N., Nazarov A.A. Infinite-Server Queueing System and Net-works. Tomsk. Publishing house STL (2015) (in Russian)Lisovskaya E., Moiseeva S., PaganoM. 325

The total capacity of customers in the MMPP/GI/∞ queueing system

PAGANO, MICHELE

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The total capacity of customers in the MMPP/GI/∞ queueing system

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