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Describe the characteristics of queuing. Name and explain some of the useful statistical models for queuing system.

• Simulation is often used in the analysis of queuing models. In a simple but typical queuing model, shown in Figure 6.1, customers arrive from time to time and join a queue (waiting line), are eventually served, and finally leave the system.

• The term "customer" refers to any type of entity that can be viewed as requesting "service" from a system.

• Therefore, many service facilities, production systems, repair and maintenance facilities, communications and computer systems, and transport and material-handling systems can be viewed as queuing systems.

The Calling Population

• The population of potential customers, referred to as the calling population, may be assumed to be finite or infinite.

• For example, consider a bank of five machines that are curing tires. After an interval of time, a machine automatically opens and must be attended by a worker who removes the tire and puts an uncured tire into the machine.

• The machines are the "customers;' who "arrive" at the instant they automatically open. The worker is the "server," who "serves" an open machine as soon as possible.

• The calling population is finite and consists of the five machines.

• In systems with a large population of potential customers, the calling population is usually assumed to be infinite. For such systems, this assumption is usually innocuous and, furthermore, it might simplify the model.

• Examples of infinite populations include the potential customers of a restaurant, bank, or other similar service facility and also very large groups of machines serviced by a technician.

  System Capacity

  • In any queuing systems, there is a limit to the number of customers that may be in the waiting line or system.

  • For example, an automatic car wash might have room for only 10 cars to waiting the line to enter the mechanism.

  • It might be too dangerous (or illegal) for cars to wait in the street. An arriving customer who finds the system full does not enter but returns immediately to the calling population.

  • Some systems, such as concert ticket sales for students, may be considered as having unlimited capacity, since there are no limits on the number of students allowed to wait 'to purchase tickets.

  • As will be seen later, when a system has limited capacity, a distinction is made between the arrival rate (i.e., the number of arrivals per time unit) and the effective arrival rate (i.e., the number who arrive and enter the system per time unit).

  The Arrival Process

  • The arrival process for infinite-population models is usually characterized in terms of interarrival times of successive customers.

  • Arrivals may occur at scheduled times or at random times. When at random times, the inter arrival times are usually characterized by a probability distribution. In addition, customers may arrive one at a time or in batches.

  • The batch may be of constant size or of random size.

  • One important application of finite population models is the machine-repair problem. The machines are the customers, and a runtime is also called time to failure.

  • When a machine fails, it "arrives" at the queuing system (the repair facility) and remains there until it is "served" (repaired).

  Queue Behavior and Queue Discipline

  • Queue behavior refers to the actions of customers while in a queue waiting for service to begin.

  • In some situations, there is a possibility that incoming customers will balk (leave when they see that the line is too long), renege (leave after being in the line when they see that the line is moving too slowly), or jockey (move from one line to another if they think they have chosen a slow line).

  • Queue discipline refers to the logical ordering of customers in a queue and determines which customer will be chosen for service when a server becomes free. OR

  • Queue discipline refers to the rule that a server uses to choose the next customer from the queue when the server completes the service of the current customer.

  • Common queue disciplines include first-in-first-out (FIFO); last-in-first-out (LIFO); service in random order (SIRO); shortest processing time first (SPT); and service according to priority (PR).

1. First in first out :This principle states that customers are served one at a time and that the customer that has been waiting the longest is served first.

2. Last in first out : This principle also serves customers one at a time, however the customer with the shortest waiting time will be served first. Also known as a stack.

3. Processor sharing: Service capacity is shared equally between customers.

4. Priority : Customers with high priority are served first.[17] Priority queues can be of two types,

5. Non-pre emptive (where a job in service cannot be interrupted) and pre emptive (where a job in service can be interrupted by a higher priority job). No work is lost in either model.

6. Shortest job first: The next job to be served is the one with the smallest size

7. Pre emptive shortest job first:The next job to be served is the one with the original smallest size.

8. Shortest remaining processing time: The next job to serve is the one with the smallest remaining processing requirement.

   Service Times and the Service Mechanism

   • The service times of successive arrivals are denoted by $S_1$, $S_2$, $S_3$, . . . They may be constant or of random duration.

   • In the latter case, {$S_1$, $S_2$, $S_3$, . . . }is usually characterized as a sequence of independent and identically distributed random variables.

   • The exponential, Weibull, gamma, lognormal and truncated normal distributions have all been used successfully as models of service times in different situations. Sometimes services are identically distributed for all customers of a given type or class or priority, whereas customers of different types might have completely different service-time distributions.

   • In addition, in some systems, service times depend upon the time of day or upon the length of the waiting line. For example, servers might work faster than usual when the waiting line is long, thus effectively reducing the service time.