A Single Service Desk Queueing System Model with Utilization and Maximum Queue Length Statistics
Classification tags: business operations management, DES, next-event time progression
Classification tags: business operations management, DES, next-event time progression
The potentially relevant object types are:
Notice that it seems preferable (more natural) to separate the service queue from the service desk and not consider the customer that is currently being served at the service desk to be part of the queue.
Conceptually, a queue is a linearly ordered collection of objects of a certain type with a First-In-First-Out policy: the next object to be removed is the first object, at the front of the queue, while additional objects are added at the end of the queue.
Potentially relevant types of events are:
Notice that a pair of start and end events, like "service start" and "service end", indicates that there is an activity that is temporally framed by these two events. It's an option to consider also activities, in addition to objects and events, in a conceptual model. We will do this in our 3rd simulation model of the service desk system.
Both object types and event types, together with their participation associations, can be visually described in a UML class diagram, as shown below.
| ON (event type) | DO (event routine) | Rule Diagram |
|---|---|---|
| customer arrival | If the service desk is busy, then the new customer queues up, else the service starts. |  |
| service start | After some time, the service ends. |  |
| service end | The served customer departs. If there are still customers waiting in the queue, then the next service starts. |  |
The involved types of events can be related with each other via their (possibly conditional) temporal succession, as visualized in the following BPMN process diagram:
In the current simulation project, the purpose of the simulation model is to compute two statistics: the service utilization and the maximum queue length. We may therefore abstract away from many of the object types from the conceptual information model:
Customer: we don't need any information about individual customers;ServiceQueue: we don't need to know who is next, it's sufficient to know the length of the queue;ServiceDesk: we don't need any information about the service desks, the information if
the service desk is busy or not is obtained by testing if the queue's length is greater than 0.ServiceClerk: we don't need any information about service clerks.Consequently, a computational design for the purpose of computing the two statistics maximum queue length
and service utilization is obtained by modeling only one object type: ServiceDesk with
one (integer-valued) attribute queueLength, and with one random variable serviceDuration()
expressed in the form of a special kind of function.
Concerning the event types described in the conceptual information model, the goal is to keep only those of them, which are really needed, in the design model. This question is closely related to the question, which types of state changes and follow-up event creation have to be modeled for being able to answer the research question(s).
In the case of the given two research questions, we need to keep track of changes of the queue length and we need to be able to add up the service durations. For keeping track of queue length changes, we need to consider all types of events that may change the queue length: customer arrivals and customer departures. For being able to add up the service durations, we need to catch service start and service end events.
After identifying the relevant event types, we can look for further simplification opportunities by analyzing their possible temporal coincidence. Clearly, we can consider customer departures to occur immediately after the corresponding service end events, without having any effects that could not be merged. Therefore, we can drop service end events, and take care of their effects (the cumulative service time computation) when handling the related customer departure event.
In addition, we can drop service start events, since they temporally coincide with customer arrivals when the queue is empty, or otherwise (when the queue is not empty) they coincide with service end (and, hence, with customer departure) events, because each service end event causes a new service start event as long as the queue is not empty.
As a result of the above considerations, we get the following two types of events:
CustomerArrival with a reference property serviceDesk with range
ServiceDesk. As an exogenous event type, CustomerArrival has a
recurrence function representing a random variable for computing the time in-between
two subsequent event occurrences.CustomerDeparture with a reference property serviceDesk with range
ServiceDesk.Notice that, for simplicity, we consider the customer that is currently being served to be part of the queue. In this way, in the simulation program, we can check if the service desk is busy by testing if the length of the queue is greater than 0.
An alternative approach would be not considering the currently served customer as
part of the queue, but rather use a Boolean attribute isBusy for being able to keep track if the
service desk ist still busy with serving a customer.
The object types and event types described above, together with their participation associations ("service desks participate in customer arrival events and in customer departure events"), are visually described in the following UML class diagram:
Notice how functions representing random variables, like serviceDuration() and recurrence(),
are marked with the keyword (or UML stereotype) «rv» standing for "random variable".
| ON (event type) | DO (event routine) |
| CustomerArrival( sd) @ t |
INCREMENT sd.queueLength IF sd.queueLength = 1 THEN SCHEDULE CustomerDeparture( sd) @ (t + sd.serviceDuration()) |
| CustomerDeparture( sd) @ t |
DECREMENT sd.queueLength IF sd.queueLength > 0 THEN SCHEDULE CustomerDeparture( sd) @ (t + sd.serviceDuration()) |
| Conceptual rule models | Rule design models |
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