# Continuous-Service M/M/1 Queuing Systems
## 1 Introduction and Motivation  
\"I wonder approximately how long I'll have to wait in line.\" This thought has gone through all our minds as we joined a queue to wait at the bank, the fast food restaurant, or any other place that required waiting for service. For the most part, we do not know the answer to this question but in queuing theory, mathematical models for queues are developed so that this and other questions may be answered. The study of queues has been rigorously carried out since the early twentieth century. [PERSON] is among the pioneer researchers in queueing theory in the early twentieth century. [PERSON] [1] studies the operations of telephone systems. The commonly used notation for queueing systems (_a/b/c_ where \(a\) describes the arrival process, \(b\) indicates the service process, and \(c\) specifies the number of servers) is first proposed by [PERSON] and [PERSON] [2, 3] writes the first textbook on the subject of queues. Applications of queueing models may be widely found in traffic control, manufacturing and production systems, healthcare, etc. Researchers often employ queueing models to study traffic flow. For example, in [4], [PERSON] and [PERSON] use an expansion method on queueing networks to study traffic control. [PERSON] utilizes queueing theory to approximate the queue length and waiting time at intersections [5, 6, 7]. Plans and scheduled transportation vehicle fleet in a congested traffic environment [8]. Queueing theory is widely applied in the study of manufacturing and production systems. For example, see [9, 10, 11] for the way queueing techniques are employed to analyze manufacturing systems. Most recently, a host of researchers actively investigate into a variation, M[X]/G(_a_, _b_)/_c_ systems, of the standard M/G/1 system. Batch arrivals and machine repairs are often imposed on these systems; hence, results obtained from their studies are most applicable to production systems; see, for example, [12, 13, 14, 15, 16, 17]. Research in effective healthcare delivery and access commonly uses queueing analysis. For example, [PERSON] employs queueing models to study waiting times in hospitals [18], and [PERSON] and [PERSON] use simulation in planning the transportation of patients to hospitals following disasters [19]. In [20], [PERSON] and [PERSON] model the utilization of beds in hospitals. On the other hand, a queueing approach is employed to manage medical appointments and to identify good nursing levels [21; 22]. The reference [23] may provide a further review on the use of queueing models in healthcare operations. The standard M/M/1 system is a queuing system where the interarrival and service times are independently and exponentially distributed with a single server. In the standard M/M/1 system, the server serves customers one by one when customers exist in the system. The server becomes idle when there are not any customers present in the system. When a customer arrives into the system during an idle period, the customer enters into the service immediately since there is no queue. However, during a busy period, a new customer arriving into the system does not preempt the current active service and hence must wait in the queue for their service. In this paper we present a novel notion of the standard M/M/1 queueing system. We call the systems under consideration the continuous-service M/M/1 queueing system. The continuous-service M/M/1 system is different from the standard system in that the server is continuously serving, and is therefore never idle. To manifest the aforementioned characteristic, we assume that there are two types of customers in this system: real customers and imaginary customers. We further assume that there are an infinite number of imaginary customers already present in the queue, but real customers arrive into the system according to a Poisson process. An arriving real customer into the system does not preempt the current service of a (real or imaginary) customer and therefore needs to join the queue.