7.3.1. Network of the [GI|M|∞]^r-type in heavy traffic

The local approach is a suitable tool to prove limit theorems provided that the input flow of the network can be defined in a constructive way. Let us consider a multichannel network of the [GI|M|∞]^r-type.

To obtain a heavy traffic limit theorem for our sequence of service processes, we need to impose some additional conditions of overloading on the network parameters as n →∞. So, service rates μ_i, i = 1, 2, ..., r, are assumed to be dependent on the series number n so that the following condition holds.

CONDITION 7. 1. – = μ_i, i = 1, 2, ..., r.

In this section, we assume that other parameters of the [GI|M|∞]^r-network do not depend on n.

For recurrent input flows ν⁽ⁱ⁾(t),i = 1, 2, ..., r, we require boundedness of the first and the second moments of interarrival times:

CONDITION 7 . 2 . – For all i = 1, 2 ,..., r

Here, we suppose the network to be empty at the initial instant:

CONDITION 7.3.- Q_i(0) = 0 for all i = 1, 2,...,r.

Under conditions 7.1–7.3 for the open [GI∣M∣∞]^r-network, we consider the sequence of stochastic processes

[7.7]

where

θ′ = (θ₁, ..., θ_r) = λ′(I − P)⁻¹ is a solution of the balance equation for the [GI|M|∞]^r-network, P(t) = exp {Δ(μ)(P − I)t} , Δ(x) = is a diagonal matrix for any vector x′ = (x₁ ,..., x_r), μ′ = (μ₁ ,..., μ_r), I = is the identity matrix.

For ξⁿ(t), the following result holds true.

THEOREM 7.1.– Let for a queueing network of the [GI∣M∣∞]^r-type conditions 7.1–7.3 be satisfied, F_i(t), i = 1, 2 ,..., r, be a non-lattice function and the spectral radius of the switching matrix P be strictly less than one. Then on any finite interval [0, T], the sequence of stochastic processes ξⁿ(t),n ≥ 1, converges weakly in the uniform topology to a diffusion process ξ(t) (ξ(0) = 0) with drift vector A(x) = A’x and diffusion matrix

where A = Δ(μ)(P − I), σ² = i = 1, 2, ...,r.

SKETCH OF THE PROOF.- Let us check conditions [7.4]-[7.6] successively.

As a sequence of monotonically increasing σ-algebras , we can take the σ-algebras generated by the random variables

where is the arrival moment of the kth call in the ith node.

As a sequence = 1, 2, ..., we take the sequence of the uniform partitions

for some α ∈ (0;1/2).

Now, for our case, condition [7.4] takes the form:

The path of the call arrived at the [GI|M|∞]^r-network through the kth node can be described by a Markov chain y^(k) (t) = {1, 2 ,..., r.r + 1} ,t ≥ 0, y^(k) (0) = k, which is defined by the infinitesimal matrix with the following entries:

The state r+ 1 for the chain y^(k) (t) is an absorbing one. Absorption in the state r+1 means that the call leaves the network. We denote by the transient probabilities of the chain, = exp (At).

Let us connect an r-dimensional process of indicator type

with the chain y^(k)(t) as follows:

where e₀ is an r-dimensional zero vector, e_j is an r-dimensional vector with its jth entry equal to one while the rest of the entries are zero.

If, at the initial instant, a call arrives to the network for its service through the ith node, then the process χ⁽ⁱ⁾(t) points out the node where the call is staying at the instant t. If χ⁽ⁱ⁾(t)=e₀, it means that the call has already been served before the instant t, and it leaves the network through the (r + 1)th node.

Denote by i = 1, ...,r, j = 1, 2,..., independent stochastic processes with the same finite-dimensional distributions as ones of χ⁽ⁱ⁾(t). Then we have the following:

[7.8]

Substituting in the form [7.8] into [7.4], we obtain:

[7.9]

Estimation [7.9] proves validity of [7.4]. Substantially, (P′(Δt_nk) − I) ξ_nk − A′ξ_nk Δt_nk has order (Δt_nk)², therefore

By virtue of the above hypothesis on the sequence of partitions, the second term on the right side of [7.9] has order n^{−(1/2−α)} and converges to zero as well. Similarly, one can check condition [7.5] relating to the diffusion matrix of the limit process.

Condition [7.6] is equivalent to Lindeberg’s condition in the central limit theorem. Thus, the proof of [7.6] can be obtained as a consequence of the following auxiliary result.

LEMMA 7.1.–(See (Anisimov and Lebedev 1992, p. 149) Let be a sequence of i.i.d. random variables with their distribution dependent on the parameter n (series number), and let , Var. Then for any A>0 and any sequence m_n → ∞ under n →∞, we have:

An effective tool to prove the convergence of functionals given for the service process of a network is Billingsley’s test.

LEMMA 7.2.– (See (Billingsley 1999, p.142) Let the sample functions of ξ (t) and ξⁿ(t), n ≥ 1, t ∈ [0,T], belong to the Skorokhod space D[0,T] with probability 1 and the finite-dimensional distributions of ξⁿ (t), n ≥ 1, t ∈ [0,T], are assumed to converge to the finite-dimensional distributions of ξ (t), t ∈ [0,T]. If for any ε > 0

[7.10]

where F(·) is a non-decreasing continuous function on [0, T], then the sequence of stochastic processes ξⁿ (t), n ≥ 1, converges to ξ (t) in the J-topology (Skorokhod’s topology).

In the framework of theorem 7.1, the convergence of ξⁿ (t) to the diffusion ξ (t) can be derived by virtue of[7.10] and Chebyshov’s inequality. Since the limit process ξ (t) has a continuous choice function with probability one, the convergence is in the uniform topology.

This completes the proof of theorem 7.1.

7.4.1. Diffusion approximation of [SM|M|∞]^r-networks

Now, let us consider a network of the [SM|M|∞]^r-type. In this model, from the outside, a common input flow arrives at the network. The flow is controlled by a semi-Markov process x(t), t ≥ 0, with the state space {1, 2 ,..., N} . It means that the calls’ arrival moments coincide with the jumps time τ_n, n = 1, 2,..., of x (t). If at τn the process x(t) jumps to the state i, then the arrived call numbered n is directed to the node j with probability h_ij∙ , . The matrix has dimension N × r. By , we denote the semi-Markov matrix of the process x(t), and by , the distribution function of the sojourn time in the state i. The service of calls and the routing algorithm are the same as in the previous model.

Note that input flows ν^{(i) (t)}, i = 1, 2, ..., r, into different nodes of the [SM|M|∞]^r -network are related.

Instead of condition 7.2 from the previous section, we assume for semi-Markov process x(t) the following:

CONDITION 7.4.– The matrix of transition probabilities of the embedded Markov chain for x(t) is irreducible, and sojourn times at each node have the first- and second-order moments:

Under condition 7.4 for F, a stationary distribution (π₁, π₂ ,..., π_N) exists, and Π denotes the matrix of the dimension N × N with N identical lines that are (π₁ ,π₂ ,..., π_N) .

The rate λ_i,i = 1, 2, ..., r, of the input flow arriving to the ith node, which is used in the centering procedure of the service process, has the form:

In order to formulate a theorem on DA for a [SM|M|∞]^r-network, the following limit correlation matrix is needed:

[7.11]

where = (h_ji ,..., h_jr) is the jth line of the matrix H, R₀ = (I − F + Π)⁻¹ − Π is the potential of the embedded Markov chain.

We obtain the following result of asymptotic analysis of the stochastic process [7.7] for the [SM|M|∞]^r-network in transient and heavy traffic regimes.

THEOREM 7.2.– Let conditions 7.1, 7.3, 7.4 be satisfied for a queueing [SM|M|∞]^r- network, and let the spectral radius of the switching matrix P be strictly less than one. Then the sequence of stochastic processes ξⁿ (t), n ≥ 1, converges weakly in the uniform topology on any finite interval [0, T] to a diffusion process ξ (t) (ξ(0) = 0) with drift vector A(x) = A’x and diffusion matrix B(t) = Δ[q′(t)A] − A′Δ[q(t)] − Δ[q(t)]A + σ², where the matrix σ² is determined via the network parameters by relations [7.11].

Theorem 7.2 can be proved using the local approach.

Note that models with similar parameters control (the [M_SM,Q|M_SM,Q|1|∞]^r-networks) are investigated under the heavy traffic conditions in Anisimov (2008, pp.164–168), using another technique.

7.4.2. Asymptotics of stationary distribution for [SM|GI|∞]^r-networks

In this section, we consider a multichannel [SM|GI|∞]^r-network. It means that service times in the jth node are i.i.d. random variables with a distribution function Gj (t ), j = 1, 2, ..., r, t ≥ 0.

At this point, we consider a service process Qⁱ (t) = t ≥ 0, i = 1, 2, ..., r, where is the number of calls in the jth node at the instant t, provided that the network is empty at the initial instant t = 0, x(0) = i, the arrival moment of the first call is the same as an exit moment τ₁ from the state i, and τ₁ is distributed in accordance with F_i(t).

Denote multivariate generating functions of the vectors Qⁱ (t),i = 1, 2 ,..., N, by Φⁱ(t, z),t ≥ 0, |z| ≤ 1.

Let L_T, 0 < T< ∞, be a complete metric space of vector functions

with components Φⁱ(t, z) determined for t ≥ 0, │z│≤ 1, measurable over t which are probability generating functions of z. A distance between Φ, Φ* ∈ L_T is equal

It can be shown that for any T > 0 the generating functions Φⁱ(t, z), 0 ≤ t ≤ T, |z| ≤ 1, are the unique solution of the following system of integral equations:

[7.12]

Under 0 ≤ t ≤ T the solution belongs to L_T.

In the right part of[7.12], are the transient probabilities of the semi-Markov process y(t) ∈ (1, 2 ,..., r, r + 1), t ≥ 0, given by the semi-Markov matrix :

Now we reference one result.

THEOREM 7.3.– (See (Lebedev 2002b)) Let the matrix F(t) be non-lattice, the matrix F = F(+∞) be irreducible and π₁,π₂ ,..., π_N be the stationary probabilities of F. If

and the spectral radius of the switching matrix P is strictly less than one, then there exists a stationary regime for the service process Q(t), and

[7.13]

For non-Poisson input flow, the stationary distribution has a complex structure. The stationary probabilities do not have a multiplicative form which is valid for the wide class of Markov stochastic networks. Properties of stationary distribution for some special cases of the [SM|GI|∞]^r-models were studied in Lebedev (2002a). It was shown that using binomial moments is an effective tool for investigating the stationary regime of such multichannel networks.

7.4.3. Convergence to Ornstein–Uhlenbeck process

In this section, we consider [SM|M|∞]^r-models. In conditions for existence of a stationary regime, we denote by Q = (Q₁ ,..., Q_r)′ a random vector corresponding to the stationary distribution. Let us proceed with the asymptotic analysis of the vector sequence Q = Qⁿ in heavy traffic for an [SM|M|∞]^r-network.

THEOREM 7.4.– Suppose conditions 7.1 and 7.4 hold for an [SM|M|∞]^r-network, F(t) is a non-lattice function, and the spectral radius of the switching matrix P is strictly less than one. Then

[7.14]

where the random vector ζ⁰ has Gaussian distribution with Eζ⁰ = 0, and the following correlation matrix:

[7.15]

The matrix σ² is given by [7.11].

A proof of theorem 7.4 is based on the functional limit theorem 7.2.

Assume that a [SM|M|∞]^r-network operates in a stationary regime while conditions 7.1 and 7.4 of overloading hold. Then the sequence

satisfies the following result.

THEOREM 7.5.– If a queueing [SM|M|∞]^r-network operates in a stationary regime and the assumptions of theorem 7.4 are satisfied, then the sequence of stochastic processes ζⁿ(t),n ≥ 1, converges weakly in the uniform topology on any finite interval [0,T] to an r-dimensional stationary diffusion Ornstein–Uhlenbeck process with the drift vector A(x) = A’x and the diffusion matrix

PROOF.– The r-dimensional Ornstein–Uhlenbeck process with the drift A(x) and the diffusion matrix B given by theorem 7.5 has stationary Gaussian distribution with zero mean and correlation matrix [7.15]. Therefore, theorem 7.4 provides weak convergence of an initial distribution for the prelimit and limit processes . Validity of other conditions of local approach can be established similarly to that in theorem 7.1.

7.5.1. Gaussian approximation of [G|M|∞]^r-networks

Now we consider a case without any restrictions imposed on the structure of the input flow ν(t) = (v⁽¹⁾(t),..., ν^(r)(t))′.

Condition 7.1 is supposed to be fulfilled for service rates μ_i,i = 1, 2, ..., r, and for the input flow we now assume that the following property holds instead of conditions 7.2 and 7.4.

CONDITION 7.5.– There exist constants λ_i ≥ 0,i = 1, 2,...,r, λ₁ + ... + λ_r = 0, such that

as n →∞, where W(t) is an r-dimensional Brownian motion with zero mean and the correlation matrix EW(1)W′(1) = σ² = symbol denotes weak convergence in the uniform topology, n is a series number.

In order to construct a limit process for the sequence of stochastic processes

where θ′ = (θ₁ ,..., θ_r) = λ′ (I − P)⁻ⁱ is a solution of the balance equation for the [G|M|∞]^r-network, λ′ = (λ₁ ,..., λ_r), we consider two independent Gaussian processes ξ⁽¹⁾(t) and ξ⁽²⁾(t) with zero means and the following correlation matrices:

The summarized process ξ⁽¹⁾ (t)+ ξ⁽²⁾ (t) describes asymptotic behavior of the sequence of stochastic processes ξⁿ(t).

THEOREM 7.6.– (See (Lebedev 2003)) Let conditions 7.1, 7.3 and 7.5 be satisfied for a [G|M|∞]^r-network, and assume the spectral radius of the switching matrix P to be strictly less than one. Then, the sequence of stochastic processes ξⁿ(t),n ≥ 1, converges weakly in the uniform topology on any finite interval [0, T] to ξ⁽¹⁾(t)+ ξ⁽²⁾(t).

The proof of theorem 7.6 is based on some auxiliary results.

LEMMA 7.3.– The finite-dimensional distributions of coincide with finite-dimensional distributions of the Gaussian process ξ⁽¹⁾(t).

This result can be proved in virtue of properties of stochastic integrals over a Gaussian process with independent increments.

Service of a call in nodes of our network is independent of other calls. In order to define the service process in a constructive way, we use the process of indicator type , t ≥ 0, i = 1,..., r, introduced in section 7.3.1.

For any natural number N and z′(k) = (z₁(k),..., z_r(k)), k = 1, 2,...,N, |z(k)| ≤ 1, we denote the joint generating function of the vectors χ⁽ⁱ⁾(t₁) ,..., χ⁽ⁱ⁾(t_N), 0 < t₁ < ...< t_N, by Φ⁽ⁱ⁾ = Φ⁽ⁱ⁾ (z(1) , ..., z(N), t₁ ,..., t_N) ,i = 1, 2, ..., r. The following formula takes place for Φ = (Φ⁽¹⁾ ,..., Φ^(r))′.

LEMMA 7.4.– For any N = 1, 2,... and 0 < t₁ < ... < t_N

where is an r-dimensional vector consisting of ones, Δt_k = t_k − t_k−1 (t₀ = 0), k = 1, 2 ,..., N.

The proof of lemma 7.4 can be derived by induction over N.

Returning to theorem 7.6, we note that its proof can be obtained from lemmas 7.3 and 7.4, and it is composed of two parts. First, convergence of finite-dimensional distributions is established. Second, convergence in the uniform topology is verified. Arguments of the proof use the method of characteristic functions and representation of the service process under fixed path of input flow as a sum of indicator processes with necessary calculations.

It should be noted that the first term of the limit process is connected with fluctuations of the input flow, and the second one with fluctuations of service time in the network nodes.

Let us prove that the limit Gaussian process in theorem 7.6 is a diffusion. With this aim in view, Markovian property of the limit process has to be proved.

7.5.2. Criterion of Markovian behavior for r-dimensional Gaussian processes

In a one-dimensional case, there is a criterion for Gaussian processes in terms of necessity and sufficiency to verify the Markovian property (see (Feller 1966, theorem 1, p. 94)). The principle condition of the criterion is written in terms of correlations and corresponds to the characteristic property of the exponential function. It is quite convenient to verify it in practice. In the multidimensional case, the situation becomes more complicated and there is no general criterion. We present a variant of the sufficient conditions for the Markovian property of r-dimensional Gaussian processes from Lebedev (2001) and apply this criterion to the limit process in theorem 7.6.

Letbe an r-dimensional Gaussian process with zero mean

and correlation matrices

THEOREM 7.7.– (See (Lebedev 2001; Livinska and Lebedev 2016)) Let for some matrix A and for all s, t (0 ≤ s < t) the functions R(s) and R(s, t) be related in the following way:

Then the r-dimensional Gaussian process ξ(t) is a Markov process, and the conditional distributions , (σ-algebra of Borel subsets of ), is Gaussian with mean vector P′(t − s)x and correlation matrix R(t) − P′(t − s)R(s)P(t − s).

Note that if G_A is the set of Gaussian processes for which the condition of theorem 7.7 is valid and the corresponding matrices are the same (equal A), then G_A satisfies the closure condition: a linear combination of two independent processes from G_A belongs G_A . Thus, we obtain the following interesting fact: the sum of two independent Markov G_A-processes is a Markov process. Note also that the multidimensional Ornstein–Uhlenbeck process satisfies the condition of theorem 7.7.

Let us apply the criterion of Markov behavior given by theorem 7.7 to the limit Gaussian process in theorem 7.6.

COROLLARY 7.1.– Let conditions of theorem 7.6 be fulfilled. Then the sequence of stochastic processes ξⁿ(t), n ≥ 1, converges weakly in the uniform topology on any finite interval [0, T] to diffusion ξ(t) (ξ(0) = 0) with the drift vector A(x) = A’x and the diffusion matrix

If we denote the corresponding correlation matrices of the process by R(t), R(s, t), s < t, then the conditions of theorem 7.7 are satisfied for them with A = Δ(μ)(P − I). So, is a Markov diffusion process. Drift vector and diffusion matrix are determined by the form of conditional distribution ,

7.5.3. Non-Markov Gaussian approximation of -networks

Now, in contrast to sections 7.5.1 and 7.5.2, the distribution function G_i(t), i = 1, 2, ..., r, of service times in the network nodes has general form and the memorylessness property can be not fulfilled.

The path of a call in such a network can be described by a semi-Markov process y(t), t ≥ 0, with the set of states {1, 2,...,r,r +1} and with its semi-Markov matrix ∣∣ defined in section 7.4.2.

In accordance with the construction, the state r +1 is an absorbing one for the process y(t). Recall that transient probabilities of the semi-Markov process y(t) are = P (y(t) = j | y(0) = i), P(t) = .

In order to define the heavy traffic transient regime for the [G|GI|∞]^r-network, we keep conditions 7.3 and 7.5 for initial distribution and for input flow, but instead of condition 7.1 we require the following.

CONDITION 7.6.– , as n → ∞, i = 1, 2,...,r,

where means weak convergence of distribution functions.

The main objective of this section is to study limit behavior of the sequence of stochastic processes

where are the transient probabilities of the semi-Markov process yⁿ(t) defined likewise y(t),t ≥ 0, with changing

In this regard, let us consider two mutually independent Gaussian processes ξ⁽¹⁾ (t) and ξ⁽²⁾ (t),t ≥ 0, with zero means and correlation matrices in the following form:

where is the mth-line of the matrix P(t), and for s < t,

If the [G|GI|∞]^r-network operates under heavy traffic transient conditions, then the next result holds for the normalized service process ξⁿ(t), n ≥ 1.

THEOREM 7.8.– Let a queueing [G|GI|∞]^r-network satisfy conditions 7.3, 7.5 and 7.6, and let the spectral radius of the switching matrix P be strictly less than one. Then the sequence of stochastic processes ξⁿ(t), n ≥ 1, converges in the uniform topology on any finite interval [0, T] to ξ⁽¹⁾(t) + ξ⁽²⁾(t).

Substantially, theorem 7.8 is an analogue to theorem 7.6. The main difference is that the limit process is a non-Markov process if the distribution function of service times is not exponential at least in one network node.

The sketch of the proof is similar as well. The statement of lemma 7.3 for the [G|GI|∞]^r-network will be the same if we consider the matrix of transient probabilities of semi-Markov process y(t) as the matrix P(t).

Going forward, we need the following result instead of lemma 7.4.

Let 0 ≤ t₁ < t₂ < ... < t_k, and

where the distribution function of sojourn time of y(t) in the initial state m is assumed to be the same as G_m(t).

Functions (t₁ , t₂ , ..., t_k) are uniquely determined by the sequence of systems of Markov renewal integral equations

LEMMA 7.5.– If the spectral radius of the matrix P is strictly less than one, then for any N = 1, 2, ... and 0 < t₁ <...< t_N we have

Like the statement of lemma 7.4, the last result can be obtained by the method of mathematical induction with respect to N.

The above models have one point in common: their parameters do not depend on time. Surely, it constricts the applications field. Especially, this point is sensitive when one simulates input flows with their real-life rates dependent on time. Further, within the framework of the simplest Markov models, we remove this restriction.

7.6.1. Gaussian approximation of -networks in heavy traffic

Let us consider a network with a non-homogeneous Poisson input flow of calls ν⁽ⁱ⁾ (t) arriving at the ith node, i = 1, 2, ..., r. Suppose Λ_i(t) is the leading function of ν⁽ⁱ⁾(t). Service times in the ith node are independent exponentially distributed random variables with rate μ_i. As before, the r-dimensional service process of calls Q(t) = (Q₁(t),..., Q_r(t))′, t ≥ 0, is studied.

The heavy traffic regime for the -network is determined by condition 7.1 and the following condition on input flow parameters.

CONDITION 7.7.– Input flows depend on n in such a way that on any finite interval

where C[0,T] is the set of continuous on [0, T] functions, the symbol stands for convergence in the uniform topology.

Let us consider two cases that are important for applications when condition 7.7 is valid. We temporarily assume that the Poisson flows ν⁽ⁱ⁾ (t), i = 1, 2 ,..., r, are regular: Λ_i(t)= , where λ_i(u) is the instant value of the parameter.

7.7a If λ_i (t) is a periodic function with period T_i, then condition 7.7 holds with

7.7b If lim_t→∞ λ_i(t) = λ_i > 0, then condition 7.7 holds with

Note, that networks with a periodic input flow are studied in more details in Livinska and Lebedev (2018).

In the context of conditions 7.1 and 7.7, we consider the sequence of stochastic processes:

are entries of the matrix

In order to describe the limit of the sequence of stochastic processes ξⁿ (t) as n → ∞, we introduce two independent Gaussian processes

The process ξ⁽¹⁾(t) is determined by zero mean:

and by the correlation matrices:

where P(τ) = exp{Δ(μ)(P − I)τ.

For the process :

The following result gives an approximation for the service process of an -network under heavy traffic conditions 7.1 and 7.7.

THEOREM 7.9.– Let conditions 7.1 and 7.7 take place for an -network, and let the network be empty at the initial instant t = 0: Q_i(0) = 0,i = 1, 2 ,..., r. Then, the sequence of stochastic processes ξⁿ (t) converges on any finite interval [0, T] as n → ∞ to the process ξ⁽¹⁾(t) + ξ⁽²⁾(t) in the uniform topology.

SKETCH OF THE PROOF OF THEOREM 7.9.– The proof follows from some auxiliary results and is similar to the proof of theorem 7.6.

In the case of non-homogeneous Poisson input flow, we have to prove the next property which is parallel to condition 7.5 supposed a priori.

LEMMA 7.6.–Let νⁿ(t) be a non-homogeneous Poisson process with the leading function Λ⁽ⁿ⁾(t) such that condition 7.7 is valid. Then the sequence of stochastic processes W⁽ⁿ⁾(t) = n ≥ 1, converges in the uniform topology on any finite interval [0, T] to a Wiener process W⁽⁰⁾(t) with EW⁽⁰⁾(t) = 0and VarW⁽⁰⁾(t) = Λ⁽⁰⁾(t).

Convergence of finite-dimensional distributions of the process W⁽ⁿ⁾ (t) to W⁽⁰⁾ (t) follows from the fact that for any natural number N and instants 0 < t₁ < ... < t_N the joint characteristic function of the distribution of νⁿ(t₁) ,..., νⁿ(t_N) is equal to

where (s(1) ,..., s(N)) ∈ ,t₀ = 0.

Now, in order to prove the convergence in the uniform topology, it is sufficient to check the following condition for any ε > 0 :

[7.16]

Based on Chebyshov inequality, we have the upper estimate:

[7.17]

On the other hand, condition 7.7 implies that

[7.18]

So, [7.16] follows from [7.17] and [7.18].

Lemma 7.6 is proved.

Throughout the section, = 1, 2,..., r, denote independent Wiener processes with zero mean and If condition 7.7 holds, they approximate the input flows ν⁽ⁱ⁾ⁿ(t).

For W^(0)′ (t) = we need the result given by lemma 7.3. Lemma 7.4 have to be satisfied as well. So, like in the case for theorem 7.6, the proof of theorem 7.9 can be obtained from lemmas 7.3, 7.4 and 7.6.

It is easily observed that the limit of ξⁿ(t) is a Markov Gaussian process. However, it can be presented as a diffusion process due to theorem 7.7.

THEOREM 7.10.– Let an -network be empty at the initial instant t = 0, let conditions 7.1 and 7.7 hold and , i = 1, 2 ,..., r. Then, the sequence of stochastic processes ξⁿ(t) ,n ≥ 1, weakly converges on any finite interval [0, T] in the uniform topology to the r-dimentional diffusion process ξ⁽⁰⁾(t) with the drift vector A(x) = A’x and the diffusion matrix

where ,

The form of the limit process as a multidimensional diffusion is attractive because a diffusion process is determined only by its local characteristics and one can use the developed tools of Markov diffusion processes for the analysis of its functionals. However, now the limit process does not reflect in detail the structure of the prelimit service process.

7.6.2. Limit process in case of asymptotically large initial load

Let us consider a case when the number of calls being served in the - network at the initial instant of time depends on n and increases as n → ∞. In this case, the service process of calls that were initially located at the network can converge to a non-zero limit, and this term can enter into the overall limit process. So, fulfillment of the following condition is required here instead of = 0, i = 1, 2, ..., r :

CONDITION 7.8.– = , i = 1, 2 ,..., r, where is a fixed vector in

, [∙] is the integer part of a number.

Here, we will consider an open -network with periodic input flow. As before, let θ′ = (θ₁ ,..., θ_r) = λ′(I − P)⁻¹ be a solution of the balance equation, where λ′ = (λ₁ ,..., λ_r), and in this case , i = 1, 2,..., r.

Next, we study the asymptotic behavior of the sequence of stochastic processes

With this in mind, we introduce in addition a Gaussian process ξ⁽³⁾(t) that does not depend on the previously introduced processes ξ⁽¹⁾(t) and ξ⁽²⁾(t). This process ξ⁽³⁾(t) is determined by the mean values

and by the correlation matrices

THEOREM 7.11.– (See (Lebedev and Livinska 2018)) Suppose that an – network with a periodical input flow has the spectral radius of its switching matrix strictly less than 1, and let conditions 7.1, 7.7a and 7.8 be satisfied. Then, the sequence of stochastic processes ηⁿ(t) converges as n → ∞ to ξ⁽¹⁾ (t) + ξ⁽²⁾ (t) + ξ⁽³⁾ (t) in the uniform topology on any finite interval [0, T].

In comparison with the statement of theorem 7.9, the additional summand ξ⁽³⁾(t) of the limit process is associated with fluctuations of service times of calls located in network nodes at the initial instant t = 0.

It is not difficult to check that for correlation matrices, we have:

So, theorem 7.11 implies the following result for ηⁿ(t).

COROLLARY 7.2.– Let for an open -network conditions 7.1, 7.7a and 7.8 be satisfied. Then the sequence of stochastic processes ηⁿ(t) converges as n → ∞ in the uniform topology on any finite interval [0, T] to an r-dimensional Ornstein-Uhlenbeck diffusion process η⁽⁰⁾ (t) in transient regime (η⁽⁰⁾(0) = η⁽⁰⁾) with the following correlation characteristics: