# Priority Multi-Server Queueing System with Heterogeneous Customers
## 2 Model Description  
We consider an \(N\) -server queueing system without a buffer. Customers of two types arrive to the system according to a \(MMAP\). The arrivals in the \(MMAP\) is directed by the underlying process \(v_{t},\ t\geq 0\), which is an irreducible continuous time Markov chain with the state space \(\{0,1,...,W\}\). In case of two types of customers, the \(MMAP\) is completely defined by the state space of underlying process and \((W+1)\times(W+1)\) matrices \(D_{k},\ k=0,1,2\), or their generating function \(D(z)=D_{0}+D_{1}z+D_{2}z^{2}\). The entries of the matrix \(D_{k}\) give the rate of transitions of the process \(v_{t},\ t\geq 0\), which are accompanied by generating a customer of type \(k,k=1,2\). Non-diagonal entries of the matrix \(D_{0}\) describe the rate of transitions of the process \(v_{t},\ t\geq 0\), which are not accompanied by generating a customer. Diagonal entries of the matrix \(D_{0}\) are negative and such that the matrix \(D(1)=D_{0}+D_{1}+D_{2}\) is an infinitesimal generator of the chain \(v_{t},\ t\geq 0\). The fundamental rate of arrivals of type-\(k\) customers is calculated as \(\lambda_{k}=\boldsymbol{\theta}\,D_{k}\boldsymbol{e}\), \(k=1,\,2\), where \(\boldsymbol{\theta}\) is a row vector of the steady state probabilities of the underlying process \(v_{t}\). The vector \(\boldsymbol{\theta}\) is the unique solution to the system \(\boldsymbol{\theta}D(1)=\boldsymbol{0},\ \boldsymbol{\theta}\boldsymbol{e}=1\). Here and in the sequal \(\boldsymbol{e}\) is a column vector consisting of \(1\)'s, and \(\boldsymbol{0}\) is a row vector consisting of \(0\)'s. The total fundamental rate of arrivals is \(\lambda=\lambda_{1}+\lambda_{2}\). The variance of inter-arrival times of customers of type \(k\) is calculated as  
\[v^{(k)}=\frac{2\boldsymbol{\theta}(-D_{0}-D_{\bar{k}})^{-1}\boldsymbol{e}}{ \lambda_{k}}-\left(\frac{1}{\lambda_{k}}\right)^{2},\ \ \bar{k}\
eq k,\ k,\bar{k}=1,2.\]  
The coefficient of correlation of lengths of two successive inter-arrival times of \(k\)-type of customers is calculated as  
\[c_{k}=\bigg{[}\frac{\boldsymbol{\theta}(D_{0}+D_{\bar{k}})^{-1}}{\lambda_{k}} D_{k}(D_{0}+D_{\bar{k}})^{-1}\boldsymbol{e}-\bigg{(}\frac{1}{\lambda_{k}} \bigg{)}^{2}\bigg{]}(v^{(k)})^{-1},\ \bar{k}\
eq k,\ k,\bar{k}=1,2.\]  
More details about a \(MMAP\) can be found in [27].  
The service time of the \(k\)-type customer has \(PH\) type distribution with irreducible representation \((\boldsymbol{\beta}_{k},S_{k})\). Here \(\boldsymbol{\beta}_{k}\) is a row vector of size \(M_{k}\), and \(S_{k}\) is a square matrix of size \(M_{k}\). Thus, the specified service time is interpreted as the time during which an underlying Markov chain \(m_{t}^{(k)},t\geq 0\), with state space \(\{1,\ldots,M_{k},M_{k}+1\}\) will reach the only absorbing state \(M_{k}+1\). The transition rates of the chain \(m_{t}^{(k)},t\geq 0\), within the space of transient states \(\{1,\ldots,M_{k}\}\) are defined by the sub-generator \(S_{k}\), and the rates of transitions to the absorbing state are defined by the vector \(\boldsymbol{S}_{0}^{(k)}=-S_{k}\boldsymbol{e}\). At the time the service starts, the state of the process \(m_{t}^{(k)},\ t\geq 0\), is selected from the state space \(\{1,\ldots,M_{k}\}\) according to the probability row vector \(\boldsymbol{\beta}_{k}\). The service rate are calculated as \(\mu_{k}=-(\boldsymbol{\beta}_{k}S_{k}^{-1}\boldsymbol{e})^{-1}\). More information about the \(PH\) type distribution can be found, e.g., in [17; 29].  
We assume that customers of type 1 have the preemptive priority. If a priority customer arrives to the system when all servers are busy and there are servers occupied with non-priority customers, then the arriving priority customer crowds out one of these customers (which is lost) and takes his/her place on the server. If all servers are occupied with priority customers, then an arriving priority customer is lost. If a non-priority customer arrives at the system when all server are busy, he/she leaves the system forever.