Evolución

Frequently queueing networks of interest are too complex to be analyzed directly, so one would like to use the steady state behavior of the limit diffusion model to approximate that of the underlying queueing system. In this chapter, we justify such an approximation procedure by studying convergence of the invariant measures for the Markov modulated queueing networks considered in Chapter 2 in heavy traffic. For simplicity, we consider an open queueing network with constant routing matrix and arrival and service rates that only depend on the state and the slowly changing Markov process (i.e., the network parameters do not depend on Xn_{). In then}th _network,

b

Qn and Yn _{will denote the normalized queue length process and the modulating Markov} process, respectively. The main result of this chapter (Theorem 4.1.1) shows that, under suitable heavy traffic and stability conditions, (Qbn, Yn) admits a stationary

distribution which converges to that of (Z, Y), where (Z, Y) is as in Chapter 3, as

n→ ∞.

networks with the following structure. Each network has K service stations each of which has an infinite capacity buffer. We denote the ith station by Pi, i ∈ IK

.

=

{1,2, . . . , K}. All customers/jobs at a station are “homogeneous” in terms of service requirement and routing decisions. Arrivals of jobs can be from outside the system and/or from internal routing. Upon completion of service at station Pi a customer is routed to some other service station or exits the system. The external arrival processes and service processes are assumed to depend on the state of the system and an auxiliary finite state Markov process. The routing mechanism is governed by a

K ×K substochastic matrix _P. Roughly speaking, the conditional probability that a job completed at station Pi is routed to station Pj equals the (i, j)th entry of the matrix _P. The above formal description is made precise in what follows.

In the nth _{network, the Markov process modulating the arrival and service rates} is denoted as {Yn_{(t) : t ≥ 0}. We assume that Y}n _{has a finite state space IH and} infinitesimal generator _Qn _{which converges to some matrix}

Q. Let Qni(t) denote the number of customers at station Pi at time t. Then the evolution of Qn can be described by the following equation

Qn_i(t) = Qn_i(0) +An_i(t)−D_in(t) + K X j=1 D_jin(t), i∈IK. (4.1.1) Here An

i(t) is the number of arrivals from outside at station Pi by time t, Din(t) is the number of service completions by time t at station Pi, and Djin(t) is the number of jobs that are routed to Pi immediately upon completion at station Pj by time t. Letting D_i0n(t) be the number of customers by time t who leave the network after service at Pi, we have D_in(t) = K X j=0 D_ijn(t). (4.1.2)

by requiring thatAn_i and Dn_ij,1≤i≤K,0≤j ≤K, are counting processes given on a suitable filtered probability space (Ωn,Fn_{, P}n_,{Fn

t }) such that for some measurable functions λn i,α˜in:IR+K×IH →IR+, the processes ˜ An_i(·)≡An_i(·)− Z · 0 λn_i(Qn(u), Yn(u))du, ˜ D_ijn(·)≡D_ijn(·)− Z · 0 Pijα˜ni(Q n (u), Yn(u))du (4.1.3)

are locally square integrable {Fn

t}martingales. Here Pi0 = 1−

PK

j=1Pij.We assume that processes An_i and D_ijn,1 ≤i ≤ K,0 ≤ j ≤ K, and Yn have no common jumps. We also require that Yn is a{Fn

t }Markov process. The functions λni and ˜αni, i∈IK, represent the arrival and service rates. We denote by IK0 (IK0 ⊆ IK) the set of indices of stations which receive arrivals from outside. In particular,λn

i(x, y) = 0 for all (x, y)∈IRK

+ ×IH whenever i∈IK\IK0. Reflecting the fact that no service occurs when the buffer is empty, ˜αn

i(x, y) = 0 if xi = 0. Let λn = (λn1, . . . , λnK)

0_{. We assume}

that, for eachi∈IK, ˜αn

i restricted to IRK+\{z ∈IRK+ :zi = 0}

×IH can be extended to a function αn_i defined on IRK₊ ×IH (that satisfies additional properties as specified below), and writeαn= (αn₁, . . . , αn_K)0. Let

bn =. λ n_−[ I−P 0 ]αn √ n .

We introduce the main assumptions on model parameters, which are similar to Assumption 2.3.1(i)-(vi).

Assumption 4.1.1.

(i) The spectral radius of _P is strictly less than 1.

(iii) For some θ2 ∈(0,∞), sup(z,y)∈IRK +×IH|b

n_{(z, y)| ≤θ} 2.

(iv) There exists a bounded Lipschitz mapb :IRK₊×IH →IRK such thatbn(√nz, y)→ b(z, y) uniformly on IRK₊ ×IH as n → ∞.

(v) There exist IRK₊-valued bounded Lipschitz functions λ, α defined on IR₊K ×IH, such that

λn₍√_{nz, y)}

n →λ(z, y),

αn₍√_{nz, y)}

n →α(z, y)

uniformly for (z, y) in compact subsets of IRK

+ ×IH as n → ∞. Furthermore,

λ= [_I−_P0]α.

(vi) For each i∈IK\IK0, there exists j ∈IK0 such that P

m ji >0 for some m∈IN. For t≥0, let b Qn(t) = Q n_(t) √ n .

From Theorem 2.3.2 in Chapter 2, it follows that under Assumption 4.1.1, asn→ ∞, (Qbn, Yn) converges weakly to a Markov process (Z, Y), where (Z, Y) is as in Definition

2.3.2. In particular, Y is a Markov process with infinitesimal generator_Q and Z is a reflected diffusion process with state dependent and Markov modulated coefficients, which can be described as follows.

Z(t) = Γ z+ Z · 0 b(Z(s), Y(s))ds+ Z · 0 σ(Z(s), Y(s))dW(s) (t), t≥0.

The driftb is as in Assumption 4.1.1(iv) and the diffusion coefficientσ is constructed as between (2.3.9) and (2.3.10). Note that b and σ satisfy Assumptions 3.2.2 and 3.2.3. Denote Φn _{≡ (}

b

Qn_{, Y}n_{),Φ≡ (Z, Y) and ϕ≡ (z, y). The following is the main} result of the chapter.

Then there exists N ∈ IN such that for any n ≥ N, the Markov process Φn admits a stationary distribution. Let πn be an arbitrary stationary distribution of Φn. Then

πn⇒π as n→ ∞, where π is as in Theorem 3.2.2.

In the following, we provide an explicit example, where assumptions of the above theorem hold.

Example 4.1.1. Let K = 2, IH = {1,2}, and _P =

   0 1₂ 1 3 0  

. The arrival and

service rate λn and αn are defined as follows. For z = (z1, z2)∈IR2+ and y∈IH,

λn(z, y) =√n(e−z1/ √ n_{+ 4) +ny,}√_n(e−z2/ √ n_{+ 4) + 2ny}0_, αn(y) = 24 5 √ ny+ 2ny,27 5 √ ny+ 3ny 0 . Therefore, bn(z, y) = e−z1/ √ n + 4−3y, e−z2/ √ n + 4−3y 0 , and b(z, y) = bn(√nz, y) = (e−z1 _{+ 4−3y, e}−z2 _{+ 4−3y)}0_{. (4.1.4)} Letq∗ = (1₄,3₄). We can construct a Markov process Yn_{, which has state space IH and}

convergent infinitesimal generator, such that it converges to a Markov processY with stationary distribution q∗. With the above model parameters, we have from Theorem 2.3.2 that (Qbn, Yn) ⇒ (Z, Y), where Z is defined as in (2.3.11) with drift b defined

as in (4.1.4) and diffusion coefficient σ constructed as between (2.3.9) and (2.3.10). We note that the constraint directions for Z are d1 = (1,−1₂)0 and d2 = (−1₃,1)0

and therefore the cone C = {−α1d1−α2d2 :α1 ≥0, α2 ≥0}. We observe that, for

z ∈IR2 +,

and the “average” drift b∗(z) = e−z1 ₋ 5 4, e −z2 ₋ 5 4 0 ∈ Co_.

In fact, for all 0< δ0 < 1₄, we have for all z ∈IR+2, b∗(z)∈ C(δ0). By Theorem 3.2.2, (Z, Y) is positive recurrent and has a unique invariant measure π. Finally, from Theorem 4.1.1, (Qbn, Yn) admits an invariant probability measure πn and πn⇒ π as

n→ ∞.