## 2 Mathematical Model of Resource Queues with Different Types of MMPP and Renewal Arrivals  
Consider a queueing system with \(n\) different customers types and assume that each customer requests a random amount of resources (see Figure 1).  
Customers arrive in the system according to a stochastic process (a Markov-modulated Poisson process (MMPP) or a renewal process). The MMPP process is given by the underlying Markov chain \(k(t)\) with a finite number of states \(K\), the set of non-negative intensities \(\lambda_{k}\). Note that \(k(t)\) is determined by the infinitesimal generator matrix \(\mathbf{Q}=|q_{\kappa k}|,\,\
u,k=1,\ldots,K\). When the Markov chain \(k(t)\) stays in the state \(k,\,k=1,\ldots,K\), the customers arrive according to the stationary Poisson process with intensity \(\lambda_{k},\,k=1,\ldots,K\). The renewal arrival process is characterized by the distribution function \(A(z)\) of the intervals between the arrivals' moments.  
At the time of occurrence of any event in the arrival process, only a single customer arrives in the system and its type is randomly selected according to the set of probabilities \(p_{i}\,(i=1,\ldots,n)\). Then the customer goes to the appropriate device type, where its service is performed during a random time interval \(\xi_{i}>0\) with the distribution function \(B_{i}(x)=P\big{\{}\xi_{i}<x\big{\}}\,(i=1,\ldots,n)\) according to the type of the customer. Moreover, depending on its type, each customer requires a random amount of resources \(v_{i}>0,\,i=1,\ldots,n\), drawn from the distribution function \(G_{i}(y)=P\{v_{i}<y\}\,(i=1,\ldots,n)\), which is independent of its service time. Let us denote by \(V_{i}(t)\) the total amount of occupied resources of the i-th type \((i=1,\ldots,n)\) at time \(t\). The goal is to find the stationary probability distribution  
Figure 1: Resource queueing system with \(n\) different customer types.  
of the n-dimensional random process \(\{V_{1}(t),V_{2}(t),\ldots,V_{n}(t)\}\). However, this process is non-Markovian; therefore, we will use the dynamic screening method for the investigation of this problem [32; 36]. Let the system be empty at moment \(t_{0}\), and let us fix any moment \(T\) in the future. The set of dynamic probabilities \(S_{i}(t)=p_{i}(1-B_{i}(T-t))\)\((i=1,\ldots,n)\) indicates that a customer arriving at time \(t\) has the \(i\) -type and it has not finished servicing at the moment \(T\) for \(t_{0}\leq t\leq T\). We will consider such customers screened.  
For resource queueing systems, we denote the amount of resources allocated by the original request that arrived at time instant \(t\) and screened by \(W_{i}(t)\) and \((i=1,\ldots,n)\). Then the probability distribution of the amount of resources allocated in the system at time instant \(T\) coincides with the probability distribution in the screened process. The main idea of the dynamic screening method is to analyze the n-dimensional process \(\{W_{1}(t),\ldots,W_{n}(t)\}\) and, substituting \(t=T\), we obtain a result for the process \(\{V_{1}(t),\ldots,V_{n}(t)\}\) at the time instant \(T\).  
It is easy to prove this property for the probability distribution of stochastic processes [34]:  
\[P\{V_{1}(T)<w_{1},\ldots,V_{n}(T)<w_{n}\}=P\{W_{1}(T)<w_{1},...,W_{n}(T)<w_{n}\} \text{, }w_{i}\geq 0\text{, }i=1,\ldots n.\]  
The resulting \(n\)-dimensional stochastic processes are also non-Markovian. Thus, we add the state of the Markov chain \(k(t)\) for a system with an MMPP-flow and a residual time from \(t\) to the next arrival \(z(t)\) for system with the renewal arrival process.