Open Markov queueing networks with control queues and quarantine node

Recently the questions of information security and using of the antivirus software are very important. Such software finds and neutralizes harmful objects and threats for systems and networks. In systems and queueing networks functioning of the similar software can be described by means of queues in which all arriving objects are checked with the subsequent neutralization of harmful objects in a quarantine. In this paper, the open queueing network with the Poisson input flow with a parameter X is investigated. Every customer goes to the node i with probability p_0i (i = 1, N, p_0i = 1). In each of N nodes there are two queues: control queue and a queue for service. Every arriving customer joins control queue, in which it is scanning, checking on compliance to requirements of the network. The times of checking have exponential distribution with a parameter v_i (i = 1, N ). After checking in the node i every customer is admitted as non-standard with probability p_t and is directed to the queue of node (N+1) immediately and independently from the other customers. Or it joins queue of standard customers in the node i with probability 1 - p. Standard customers are served by the device, service times in each node are independent and have exponential distribution with parameter ц (i = 1, N ). After service at the node i every customer is sent immediately to the node j with probability p_ij, and it leaves the network with probability p_i0 (l ¹N=₁ p_tj + p_i0 = 1, i = 1, N) . In the node (N+1) called quarantine the restoration of qualities of non-standard customers or their neutralization is carried out. The times of such service have exponential distribution with a parameter ^_n+₁. After service at the node (N+1) every customer is sent immediately to the node j with probability p_N+y, and it leaves the network with probability p_N+₁₀ (l 1=₁ P_N+₁ j + P_N+₁₀ = 1) . Network state at the moment t is characterized by a vector x(t) = (X1(t), *2(t),. ·., Xn+1(t)), where x_i(t) = (m(t), n_i(t)) is the state of the node i at the moment t, m_i(t) is the number of the customers in the control queue, n_i(t) is the number of the customers in the queue for service in the node i at the time moment t (i = 1, N ); x_N+₁(t) = n_N+₁(t) is the number of the customers in the quarantine node (N+1) at the moment t. The process x(t) is homogeneous Markov process with continuous time and finite or countable states spaceX = X₁ xX₂x... xX_N+_b где Xi = {х,- = (mi, n): m_h щ > 0, i = 1,N }, Xn+1 = {xn+1 = nn+1: nn+1 > 0}. Traffic equations are following ^si = p0i + I ^s j⁽¹ - pj⁾pji + sn+1pn+1i^, i = ¹N, j=¹ N ^sN+1 = I ^s jpj. j=1 Let {p(x), xe X} be stationary distribution of states probabilities for the process x(t). We proved the following statement. If for all i = 1, N inequalities }£± < ₁ tej⁽¹ - p, ⁾ < ₁ ^N+1 < ₁ are carried out, then Markov process x(t) is ergodic and stationary distribution of the network states probabilities has the following form p(x) = p1(x1) p2(x2). pn+1(xn+1), x e X, where _Pl (x) = [ ^ J' [ X^L-p) J - ^ - ], i = 11, ^xs N+11 ⁿ+1 L _-^xs N+11 ^ц N+1 j | ^ц N+1 j (s,-, 1, N +1) is solutions of the traffic equations.

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