**Queueing Model:**

Consider $n$ independent, parallel $M/M/1$ queues with identical arrival rate $\lambda$ and service rate $\mu$. For each $M/M/1$ queue, we use the FCFS (First Come First Served) discipline and *if there is some customer in service, no more customers can enter it*.            

For each customer $c$, its start time, finish time, service interval are denoted by $c_{.st}$, $c_{.ft}$, and $[c_{.st}, c_{.ft}]$, respectively.

**Problem:**

I want to study some concurrency-related problems in such queueing system in the long run. 

> (1). Given two different $M/M/1$ queues $Q_i$ and $Q_j$ and a customer $c$ served by $Q_i$, what is the probability that it *starts* during the service interval of some customer $c'$ served by $Q_j$ (i.e., $c_{.st} \in [c'_{.st}, c'_{.ft}]$)?

Notice that $c'$ will be unique if $c_{.st} \in [c'_{.st}, c'_{.ft}]$ holds (see the figure below).

> (2). Given a customer $c'$ served by $Q_j$, what is the probability that there are exactly $m$ customers each of which (denoted $c''$) *finishes* during the service interval of $c'$ (i.e., $c''_{.ft} \in [c'_{.st}, c'_{.ft}]$)?

Notice that there may be more than one customer in $Q_k$ ($Q_k \neq Q_j$) satisfying the condition $c''_{.ft} \in [c'_{.st}, c'_{.ft}]$ (see the figure below).

> (3) *Combine problems (1) and (2):*          
Given two different $M/M/1$ queues $Q_i$ and $Q_j$ and a customer $c$ served by $Q_i$, let $c'$ be the customer served by $Q_j$ satisfying $c_{.st} \in [c'_{.st}, c'_{.ft}]$ (i.e., $c$ *starts* during the service interval of $c'$).      
The set of customers that finishes before $c$ starts is denoted by $c^{\prec} = \{c'': c''_{.ft} \le c_{.st} \}$.        
What is the probability that there are **exactly $k$ customers in $c^{\prec}$** each of which (denoted by $c''$) *finishes* during the service interval of $c'$ (i.e., $c''_{.ft} \in [c'_{.st}, c'_{.ft}]$)?

![parallelmm1_threeproblems][1]

> In addition, are such *concurrency-related problems* are the typical ones studied in the literature on queueing theory? Any references related to similar problems are also well appreciated.


  [1]: http://i1.tietuku.com/5eeddcb2557fb93c.png