# Continuous-Service M/M/1 Queuing Systems
## 2 Derivation of the Steady-State Distribution  
This section seeks to derive the steady-state distribution of the continuous-service M/M/1 queueing system. The model of our continuous-service M/M/1 systems and that of the standard M/M/1 system share many similarities. In the meantime, they do differ greatly in many aspects. Figure 1 depicts the birth-death model of our systems. We describe our notation in the following. The state of a system refers to the number of real customers residing in the system. As shown in Figure 1, a system may be in a positive state or a negative state. Most notably, a system is in a positive state if the server is currently serving a real customer. On the contrary, a system is in a non-positive state if the server is currently serving an imaginary customer. More specifically, let \(N\) represent the number of real customers in a system. Then, that \(N=0\) means that there is no real customers in the system; note, however, that the server must currently be serving an imaginary customer (infinitely many imaginary customers in system to ensure continuous-service of the server upon emptying of all real customers in system). When \(N=1\), this means that there is one real customer in the system with 0 in the queue and 1 in service; when \(N=-1\), this, on the other hand, means that there is 1 real customer in the system with the 1 in queue while an imaginary customer currently at service (a newly arriving real customer into the system does not preempt the current service of an imaginary customer and hence must wait in queue). If \(N=2\), this means that there are 2 real customers in the system with 1 in queue and 1 in service; if \(N=-2\), this means that there are 2 real customers in the system with the 2 in queue while an imaginary customer currently at service. In general, that \(N=j\) indicates that there are \(j\) real customers in the system with \(j-1\) in queue and 1 in service, and that \(N=-j\) implies that there are \(j\) real customers in the system with the \(j\) in queue while an imaginary customer is currently at service, \(j=1,2,\ldots\). The state \(N=0\) simply stands for the fact that there is not any real customer in the system but an imaginary customer is currently being served. Subsequently, we let \(\pi_{j}=P(N=j)\), the probability that the system is in state \(j\), \(j=\ldots\), \(-2\), \(-1\), \(0\), \(1\), \(2\), \(\ldots\) ; in other words, \(\pi_{j}\) is the probability that there are \(j\) real customers present in the system accordingly. In addition, the following are also being employed:  
\(\lambda\): Exponential arrival rate of real customers into the system,  
\(\mu\): Exponential service rate of the server for all customers, real and imaginary,  
\(\rho\): Traffic intensity of the system with \(\rho=\frac{\lambda}{\mu}\) and \(0\leq\rho<1\)[24]. We now derive the steady-state distribution \(\pi=\{\ldots,\pi_{-2},\pi_{-1},\pi_{0},\pi_{1},\pi_{2},\ldots\}\) for our system. To this end, we work out the flow balance equations by making a cut as shown in Figure 2 and employing the notion of flow balance where net flow into a cut must equal net flow out of the cut [24]. We now proceed to write the flow balance equations. For the cuts on the left side of State 0, we have:  
\[(\text{Between States 0 and }-1):\;\lambda\pi_{0}=\mu(\pi_{-1}+\pi_{-2}+\pi_{-3}+\cdots)\]  
as shown in Figure 2.  
Subsequently,  
\[(\text{Between States }-1\text{ and }-2):\;\lambda\pi_{-1}=\mu(\pi_{-2}+\pi_{-3}+ \pi_{-4}+\cdots)\]  
Figure 1: A rate diagram for the continuous-service M/M/1 queueing system.