SIMULATION OF QUEUEING SYSTEMS

Simulation of Queueing SystemsA queueing system is described by itsCalling populationNature of arrivalsService mechanismSystem capacityQueueing discipline

Single channel queueIn single channel queue,The calling population is infinite; that is if a unit leaves the calling population and joins the waiting line or enters service, there is no change in arrival rate of other units that may need service.Arrivals for service occur one at a time in a random fashion; once they join the waiting line, they are eventually served. Service times are of random length according to a probability distribution which does not change over time.The system capacity has no limit, meaning that any number of units can wait in line.Finally, the units are served in the order of their arrival (FIFO- First In First Out) by a single server or channel.

Single channel queue (Contd.,)In single channel queue,Arrivals and services are defined by the distribution of the time between arrivals and the distribution of service times, respectively.For any simple single or multi-channel queue, the overall effective arrival rate must be less than the total service rate, or the waiting line will grow without bound. They are termed as explosive or unstable.The state of the system is the number of units in the system and the status of the server, busy or idle.An event is a set of circumstances that cause an instantaneous change in the state of the system.The two possible events that can affect the state of the system are: entry of a unit into the system (the arrival event) or the completion of service on a unit (the departure event).The queueing system includes server, the unit being served and the units in the queue.The simulation clock is used to track simulated time.

Single channel queue (Contd.,)Service-just-completed flow diagram

Single channel queue (Contd.,)Unit-entering-system flow diagram

Single channel queue (Contd.,)Potential unit acts upon arrival

Server outcomes after service completion

Single channel queue (Contd.,)Interarrival and Clock times

Single channel queue (Contd.,)Service times

Single channel queue (Contd.,)Simulation Table Emphasizing Clock Times

Single channel queue (Contd.,)Chronological Ordering of Events

Single channel queue (Contd.,)Number of customers in the system

Single channel queue (Contd.,)A small grocery store has only one checkout counter. Customers arrive at this checkout counter at random from 1 to 8 minutes apart. Each possible value of interarrival time has the same probability of occurrence, as shown in table 1. The service times vary from 1 to 6 minutes with the probabilities shown in table 2. The problem is to analyze the system by simulating the arrival and service of 20 customers.

Single channel queue (Contd.,)Distribution of time between arrivals

Single channel queue (Contd.,)

ACustomerBTime between arrivals (minutes)CArrival timeDService time(Minutes)ETime service begins

FTime customer waits in QueueGTime service endsHTime customer spends in systemIIdle time of server-

Single channel queue (Contd.,)Arrival time = cumulative of time between arrivalsTime service begins = Max (Arrival time of customer, (Service time of previous customer + Time service begins of previous customer))Time customer waits in queue = Time service begins Arrival timeTime service ends = Time service begins + Service timeTime customer spends in system = Time customer waits in queue + Service timeIdle time of server = Time service begins for customer Time service ends for previous customer

Single channel queue (Contd.,)

Single channel queue (Contd.,)

Single channel queue (Contd.,)

Expected service time (probability distribution is different)Expected time between arrivals (Probability distribution is same)E(S) Expected service times Service timep(s) Probability distribution of service time E(A) Expected time between arrivalsa lowest value of time between arrivalsb highest value of time between arrivals