# Single Server And Multi Server Waiting Line Models

von Phillzz · 05.07.2017

We will assume arrivals are on an individual basis. The difference is best illustrated by the arrival of a car to a parking lot at a restaurant. One driver leaving the car to enter the restaurant would represent the arrival of one unit or customer to the waiting line system. If a bus pulls in, there could be a batch arrival of 30 customers. Did you ever notice that the bus stalls are behind the Cracker Barrel Restaurants on the interstate highways - just so you can't see all those batch arrivals before you pull off!

It is also assumed that the arrivals are nonscheduled, and the arrival of one unit is independent of, or does not impact, the arrival of other units. Whenever these assumptions are made, arrivals are assumed to follow the Poisson Probability Distribution, a member of the family of discrete probability distributions.

The Poisson Probability Distribution is completely described by its mean, which is given the Greek symbol lambda. In a waiting line system, the mean we are referring to is the mean arrival rate. For example, we may say that the mean arrival rate is 4 calls per hour to a catalog company's telephone bank. Another way of representing the mean arrival rate is to take its inverse, which gives us the mean time between arrivals.

The Exponential Distribution is used to model the probabilities of continuous variables such as time, in the case of waiting line systems. The Greek Symbol Mu, is used to describe the mean of the Exponential Distribution. If you have the mean time between arrivals, you can find the mean arrival rate by the similar procedure - taking the inverse. For example, what if we knew that the average time between arrivals to a bank teller was 5 minutes.

The mean arrival rate would be computed as follows: Waiting Line Configuration Waiting lines may be infinite or truncated. For all of the models we will examine except one, we will assume infinite line length. One of the models we will examine is designed to model situations where no waiting is allowed - the ultimate of truncated systems. My telephone allows no waiting - if I am talking to someone, the next caller gets a busy signal.

However, airline reservation systems allow callers to a busy reservation agent to wait in a queue. The word service channel is used rather than server to avoid confusion. A single service channel may have many servers, but room for only one customer. That is called a single service channel. Queuing theory provides tools needed for analysis of systems of congestion. Mathematically, systems of congestion appear in many diverse and complicated ways and can vary in extent and complexity.

A waiting line system or queuing system is defined by two important elements: The customer population can be considered as finite or infinite. The customer population is finite when the number of customers affects potential new customers for the service system already in the system. When the number of customers waiting in line does not significantly affect the rate at which the population generates new customers, the customer population is considered infinite. Customer behavior can change and depends on waiting line characteristics.

In addition to waiting, a customer can choose other alternative. When customer enters the waiting line but leaves before being serviced, process is called Reneging. When customer changes one line to another to reduce wait time, process is called Jockeying. Balking occurs when customer do not enter waiting line but decides to come back latter.

Another element of queuing system is service system. The number of waiting lines, the number of servers, the arrangements of the servers, the arrival and service patterns, and the service priority rules characterize the service system. Queue system can have channels or multiple waiting lines. Examples of single waiting line are bank counter, airline counters, restaurants, amusement parks. In these examples multiple servers might serve customers. In the single line multiple servers has better performance in terms of waiting times and eliminates jockeying behavior than the system with a single line for each server.

System serving capacity is a function of the number of service facilities and server proficiency. In queuing system, the terms server and channel are used interchangeably. Queuing systems are either single server or multiple servers. Single server examples include gas station food mart with single checkout counter, a theater with a single person selling tickets and controlling admission into the show. Multiple server examples include gas stations with multiple gas pumps, grocery stores with multiple cashiers, multiple tellers in a bank.

Services require a single activity or services of activities called phases. In a single-phase system, the service is completed all at once, such as a bank transaction or grocery store checkout counter. In a multiphase system, the service is completed in a series of phases, such as at fast-food restaurant with ordering, pay, and pick-up windows. Queuing system is characterized by rate at which customers arrive and served by service system.

Arrival rate specifies the average number of customers per time period. The service rate specifies the average number customers that can be serviced during a time period.