Joint stationary distribution of $m$ queues in the $N$-server queueing system with reordering

The paper considers a continuous-time $N$-server queueing system with a buffer of infinite capacity and customer reordering. The Poisson flow of customers arrives at the system. Service times of customers at each server are exponentially distributed with the same parameter. Each customer obtains a sequential number upon arrival. The order of customers upon arrival should be preserved upon departure from the system. Customers which violated the order form different queues in the reordering buffer which has infinite capacity. If there are $n$, $n=\overline{1,N}$, customers in servers, then the latest customer to occupy a server is called the $1$st level customer, the last but one — the $2$nd level customer, $\ldots$, the first — the $n$th level customer. Customers in the reordering buffer that arrived between the $1$st level and the $2$nd level customers, form the queue number one. Customers, which arrived between the $2$nd level and the $3$rd level customers, form the queue number two, etc. Customers, which arrived between the $N$th level and the $(N-1)$th level customers, form the queue number $(N-1)$ in the reordering buffer. Mathematical relations in terms of $Z$-transform, which make it possible to calculate the joint stationary distribution of the number of customers in the buffer, servers, and in the $1$st, $2$nd, $\ldots$, $m$th queues ($m=\overline{1,N-1}$) in the reordering buffer, are obtained.