Key words: Markov-modulated queues, waiting time, heavy traffic.Markov-modulated queueing systems are those in which the input process or service mechanism is influenced by an underlying Markov chain. Several models for such systems have been investigated. In this paper we present heavy traffic analysis of single queueing system with Poisson arrival process whose arrival rate is a function of the state of Markov chain and service times depend on the state of the same Markov chain at the epoch of arrivals

Dimitrov, Mitko

Bulgarian Digital Mathematics Library at IMI-BAS

Pliska Stud. Math. Bulgar. 20 (2011), 53–62STUDIA MATHEMATICABULGARICASINGLE-SERVER QUEUEING SYSTEM WITHMARKOV-MODULATED ARRIVALS AND SERVICE TIMESMitko DimitrovMarkov-modulated queueing systems are those in which the input processor service mechanism is influenced by an underlying Markov chain. Severalmodels for such systems have been investigated .In this paper we present heavy traffic analysis of single queueing systemwith Poisson arrival process whose arrival rate is a function of the state ofMarkov chain and service times depend on the state of the same Markovchain at the epoch of arrivals.1. IntroductionMarkov process is natural random environment and is extensively used for model-ing the arrival rate and service times in queueing theory and many other applica-tions. The presence of this Markov process is attributed to stochastic variationsor heterogeneity of arrivals and service and actually it is a component of thequeueing system considered.Significant effort is currently being devoted to the development of integratedcommunication systems, which can support a wide range applications includingvoice, video and data.There are many exact analytical results, computational techniques and ap-proximations of the basic characteristics of Markov-modulated queueing model2000 Mathematics Subject Classification: 60K25.Key words: Markov-modulated queues, waiting time, heavy traffic.54 Mitko Dimitrov[1], [2], [3], [4], [5]. The asymptotic behavior of performance characteristics ofsuch queues are derived in some limit cases as heavy traffic. Heavy traffic analy-sis of Markov-modulated queueing models is suggested by Burman and Smith [2]where an M/G/1 type queue with Markov-modulated arrivals was investigated.Genadi Falin and Anatoli Falin [3] suggested another approach to the same prob-lem. It is based on certain “semi-explicit” formula for the stationary distributionof the virtual waiting time and its mean value in heavy traffic assumptions. Weapplied this approach to single queueing system with arrival rate and service timedepending on the state of Markov chain at the arrival epoch.2. Some preliminary resultsLet us consider an irreducible continuous time Markov chain Z(t) with finitestate space S = {0, 1, . . . ,K}, transition rates αnm, αn = −αnn and stationarydistribution pi = (pi0, pi1, . . . , piK). The infinitesimal matrix of the process isdenoted as Q.The linear equationsQ(x0, x1, . . . , xK)T = (b0, b1, . . . , bK)Thas a solution if the vector (b0, b1, . . . , bK) is orthogonal to the stationary vector pi.Now let f(n) be a nonnegative function defined on the state space of Z(t),Hn(x)is the probability distribution with two moments hn1, hn2, n = 0, 1, . . . ,K andf =K∑n=0pinf(n)hn,1.Then the linear algebraic equations(1)K∑m=0αnmxm = f − f(n)hn1, 0 ≤ n ≤ Kwith unknowns x0, x1, . . . , xK always has a solution.Since the Markov chain Z(t) is finite and irreducible, it is also ergodic chain,and pij > 0 for j = 0, 1, 2, . . . ,K. Then the stationary distribution pi = (pi0,pi1, . . . , piK) can be found from the Kolmogorov equations piQ = 0 and normal-ization conditionK∑n=0pin = 1. In coordinate form Kolmogorov equations arepi0α0j +K∑i=0piiαij = 0, 0 ≤ j ≤ K.Single-Server Queueing System with Markov-Modulated Arrivals. . . 55If we consider the equationspi0α0j +K∑i=0piiαij = 0, 0 ≤ j ≤ K,then pi0(α01, α02, . . . , α0K) = −(pi1, . . . , piK)Awhere A =α11 α12 . . . α1K...αK1 αK2 . . . αKK, and detA 6= 0.Then (α01, . . . , α0K)A−1 = −(pi1, . . . , piK)pi0Now, if we define matrix R as R =0 0 . . . 00... A−10, then matrix QR isequal to(2)0−pi1pi0−pi2pi0. . .−piKpi00 1 0 . . . 00 0 1 0 . . .. . . . . . . . . . . . . . .0 0 . . . . . . 1and for any vector y = (y0, y1, . . . , yK) we have yQR = y −y0pi0pi.It is obvious that vectorx = (x0, x1, . . . , xK)T = R(f − f(0)h01, f − f(1)h01, . . . , f − f(K)hK1)is a solution to the linear equations (1).3. An M/G/1 queueing system with Markov modulated arrivalsand service timesWe make the following assumptions1. The modulating Markov chain Z(t) is exactly as in the Section 2.2. During a time-interval in which Z(t) = n, customers arrive in a Poisson56 Mitko Dimitrovprocess at a rate λf(n)(0 < f(n) <∞) and then service times have a distributionHn. Given the states of the Markov chain, the service times are conditionallyindependent and also independent of the arrival process.3. There is a single server and the queue discipline is first come, first served.We denote by W (t) the virtual waiting time. The system is ergodic if theinequality ρ =K∑n=0λf(n)hn1pin < 1 holds. From now on we assume that ρ < 1and system is in steady state. The presence of the Markov chain Z(t) implies thatwe need to designate the waiting time process as {W,Z} = {W (t), Z(t), t ≥ 0}.Given the state of Z(t) = n, the probability that a customer arrives during (t, t+dt] and make a service demand not exceeding x is given by λf(n)Hj(x)dt+0(dt).As in the standard model it is seen that (W,Z) is a time-homogeneous Markovprocess on the state space R+xS. To derive integro-differential equations forF (x, t) = P{W (t) < x,Z(t) = n} we process as on the standard model but notethe presence of the modeling chain Z(t). Thus considering the process over theintervals (0, t],(t, t + dt] we find thatFn(x, t+ dt) = Fn(x+ dt, t)(1 − λf(n)dt+ αnndt)++∞∫0Fn(x+ dt− ν, t)λf(n)dHn(ν) +∑m6=nαmnFm(x, t)dt.This leads toF ′n(x) = (λf(n) + αn)Fn(x) + λf(n)x∫0Fn(x− y)dHn(y) +∑m6=nαmnFm(x).LetPn0 = P{W (t) = 0, Z(t) = n}ϕn(s) = E{e−sW (t);Z(t) = n} = Pn0 +∞∫0+e−sxdFn(x)hn(s) =∞∫0e−sxdHn(x).Single-Server Queueing System with Markov-Modulated Arrivals. . . 57In terms of Laplace transform we obtain for ϕn(s), n = 0, 1, . . . ,K the nextequations(3)K∑m=0ϕm(s)αmn = [λf(n)(1− hn(s)− s)]ϕn(s) + sPn0.FromK∑n=0αmn = 0 and summing both sides of (3) we get(4)K∑n=0Pn0 =K∑n=0[1− λf(n)1− hn(s)s]ϕn(s).If s→ 0 the sumK∑n=0[1− λf(n)1− hn(s)s]ϕn(s) tends toK∑n=0[1− λf(n)hn,1]pin = 1−K∑n=0λf(n)hn1pin = 1− ρandK∑n=0Pn0 = 1− ρ. Taking the first derivative at the point s = 0 from the bothsides of (4) we have(5)K∑n=0(1− λf(n)hn1)Wn =K∑n=0λf(n)hn22pin.Now multiplying both sides of (3) by xn and summing with respect to n wegetK∑n=0K∑m=0ϕm(s)αmnxn =K∑n=0[λf(n) [1− hn(s)]− s]ϕn(s)xn + sK∑n=0Pn0xn,orK∑m=0ϕm(s)(f − f(m)hm1) =K∑n=0xn [λf(n)(1− hns)− s]ϕn(s) + sK∑n=0Pn0xn.Differentiating this equation with respect to s at the point s = 0 we have(6) −K∑m=0Wm (f − f(m)hm1) =K∑n=0xnpin(λf(n)hn1 − 1) +K∑n=0Pn0xn.58 Mitko DimitrovMultiplying both sides of (6) with λ and summing (5) and (6) we get theequationK∑m=0Wm(1− ρ) =K∑n=0λf(n)hn22pin +K∑n=0λxnpin(λf(n)hn1 − 1) +K∑n=0λPn0xn.Thus we obtain the main result.Theorem 1. The mean virtual waiting time is given by the formulaW =K∑n=0λf(n)pinhn22+K∑n=0λxnpin (λf(n)hn1 − 1) +K∑n=0λPn0xn1− ρwhere the variables xn are a solution of the linear algebraic equations (1), andthe probabilities Pn0 satisfy equationK∑n=0Pn0 = 1− ρ.Our goal in this paper is to investigate the queue in heavy traffic condition.If ρ = λK∑n=0f(n)hn1pin → 1 then all probabilities Pn0, n = 0, 1, . . . ,K tendsto 0 becauseK∑n=0Pn0 = 1− ρ→ 0 as ρ→ 1.Thus we prove the next theorem.Theorem 2. If ρ→ 1 (under heavy traffic)(1− ρ)W =K∑n=0xnf(n)pinhn22f+K∑n=0xnpin(f(n)hn1 − f)f2From Pn0 > 0, n = 0, 1, . . . ,K,K∑n=0Pn0 = 1− ρ follows that:K∑n=0Pn0xn ∈ (1− ρ)(x∗, x∗), x∗ = minn{xn}, and x∗ = maxn{xn},and the next theorem is proved.Single-Server Queueing System with Markov-Modulated Arrivals. . . 59Theorem 3. Let x∗ = minn{xn}, x∗ = maxn{xn}.Thenλx∗ ≤W −K∑n=0λxnf(n)pinhn22+K∑n=0λxnpin(λf(n)hn1 − 1)1− ρ≤ λx∗We shall consider stationary distribution of the process (W (t), Z(t)) as ρ→ 1.Theorem 4.a) random variables W (t) and Z(t) are asymptotically independent;b) the random variable (1− ρ)W (t) is asymptotically exponential with the meanK∑n=0pinhn2f(n)2f+K∑n=0pin(hn1f(n)− f)xnf2.P r o o f. Let us consider several notations as ϕ(s) = (ϕ0(s), ϕ1(s), . . . , ϕK(s)),P 0 = (P00, . . . , PK0) and diagonal matrices F = diag (f(0), f(1), . . . , f(K)),H˜(s) = diag (1− h0(s), 1− h1(s), . . . , 1− hK(s)) , H˜1 = diag(h01, h11, h21, . . . ,hK1) and H˜2 = diag(h02, h12, . . . , hK2), E = diag (1, 1, . . . , 1).In this notations we can rewrite equation (3) as(7) ϕ¯(s)Q = ϕ¯(s)[λH˜(s)F − sE]+ sP¯0Multiplying from the right both sides of (7) with e¯ and P¯0e¯ = 1 − ρ ≡ ε weget(8) ϕ¯(s)[λH˜(s)F − sE]e¯T + εs = 0From equation (2) multiplying again both sides of (7) from the right side by thematrix R we haveϕ¯(s)QR = ϕ¯(s)[λH˜(s)F − sE]R+ sP¯0Ror(9) ϕ¯(s) = ϕ0(s)pipi0+ ϕ¯(s)[λH˜(s)F − sE]R+ sP¯0R60 Mitko DimitrovNow, let us replace vector ϕ¯(s) from (9) to the right side of (9), and aftersome algebraic transformation we get the equationϕ¯(s) = ϕ0(s)pipi0[E + [λH˜(s)F − sE]R]+(10) +s2ϕ¯(s)[[λH˜(s)sF − E]R]2+ sP¯0R[E + [λH˜(s)F − sE]R]== ϕ0(s)pipi0[E + λH˜(s)FR− sR] + sY¯ (s)whereY¯ (s) = sϕ¯(s)[[λH˜(s)sF − E]R]2+ P¯0R[E + [λH˜(s)F − sE]R]Substituting ϕ¯(s) = ϕ0(s)pipi0[E + λH˜(s)FR− sR] + sY¯ (s) into (8) we,ll findan expression for ϕ¯0(s) .Really[ϕ0(s)pipi0[E + λH˜(s)FR− sR] + sY¯ (s)].[λH˜(s)F − sE]e¯T + εs ==ϕ0(s)pi0[pi[λH˜(s)F − sE]e¯T + pi[λH˜(s)F − sE]R[λH˜(s)F − sE]e¯T]++sY (s)[λH˜(s)F − sE]e¯T + εs =ϕ0(s)pi0(B1(s) +B2(s)) +A(s) = 0whereA(s) = εs+ sY¯ (s)[λH˜(s)F − sE]e¯T ,B1(s) = pi[λH˜(s)F − sE]e¯TB2(s) = pi[λH˜(s)F − sE]R[λH˜(s)F − sE]e¯TNow, replace s by εs and let λ increase to 1/f as ε→ 0. Finally we obtainA(εs) = ε2s+ ϕ2s2Y¯ (εs)[λH˜(εs)Fεs− E]e¯T = ε2s+ o(ε2),Single-Server Queueing System with Markov-Modulated Arrivals. . . 61because Y¯ (εs)→ 0 as ε→ 0.B1(εs) = pi[λH˜(εs)F − λH˜1Fεs+ λH˜1Fεs − εsE]e¯T == −εs(pie¯T − λpiH˜1F e¯T)− λpiH˜1εs − H˜(εs)ε2s2ε2s2F e¯T == −εs(1− ρ)−λpiH˜2F e¯T2ε2s2 + o(ε2) = −ε2s−piH˜2F e¯T2fε2s2 + o(ε2),B2(εs) = piε2s2[λH˜(εs)εsF −E]R[λH˜(εs)εsF − E]e¯T == piε2s2[λH˜1F − E]R[λH˜1F − E]e¯T + o(ε2) == −piε2s2f2[H˜1F − f ]R[f e¯T − H˜1F e¯T ] + o(ε2)becauseH˜(εs)εs→ H˜1,H˜1εs− H˜(εs)ε2s2→H˜22.Finally we getϕ0(εs) = pi0ε2s+ o(ε2)ε2s+piH˜2F e¯T ε2s22f+pif2[H1E − fE]R[f e¯T − H˜1F e¯T]ε2s2 + o(ε2)andlimε→0ϕ¯(εs) = limε→0ϕ0(εs)pipi0=pi1 +M.s,whereM =K∑n=0pinf(n)hn22f+1f2K∑n=0pin(hn1f(n)− f)xn,sinceR[fE − H˜1F ]e¯T = x¯ = (x0, x1, . . . , xK). REFERENCES[1] F. Baccelli, A. M. Makowski. Martingale relations for the M/G/1queue with Markov -modulated Poisson input. Stochastic Processes Appl.,38 (1991), V. 2, 99–133.62 Mitko Dimitrov[2] D. Y. Burman, D. R. Smith. An asymptotic analysis of a queueing systemwith Markov-modulated arrivals. Operations Research, 34 (1986), No 1, 105–119.[3] G. I. Falin, A. I. Falin. Heavy traffic analysis of M/G/1type queueingsystems with Markov-modulated arrivals. TOP, 7 (1999), No 2, 279–291.[4] W. Fisher, K. Hellstern. The Markov-modulated poisson process(MMPP) cook-book. Performance Evaluation, 18 (1992), 149–171.[5] N. U. Prabhu,Y. Zhu. Markov modulated queueing systems. Queueingsystems Theory Appl., 5 (1989), 215–245.Mitko DimitrovUniversity of National and World EconomyStudent town, Hristo Botev8, December Str.1700 Sofia, Bulgariae-mail: mcdimitrov@abv.bg

Single-Server Queueing System with Markov-Modulated Arrivals and Service Times

http://sci-gems.math.bas.bg/jspui/bitstream/10525/2206/1/Pliska-20-2011-053-062.pdf

Single-Server Queueing System with Markov-Modulated Arrivals and Service Times

Abstract

Similar works

Full text

Available Versions

Bulgarian Digital Mathematics Library at IMI-BAS