Queuing Theory Study Notes for Mechanical Engineering

Queuing Theory

Queuing Theory: Queuing models are used to predict the performance of service systems when there is uncertainty in arrival and service times.

The simplest possible (single stage) queuing systems have the following components: customers, servers, and a waiting area (queue).

An arriving customer is placed in the queue until a server is available. To model such a system we need to specify:

Single stage queuing system: In this note, we will always assume that customers are served in the order in which they arrive in the system (First-Come-First-Served or FCFS). For the characteristics of the arrival and service processes we will make various assumptions, and in general, queuing models are classified according to the specific assumptions made.

Some of the following basic queuing modes are:

• M/M/s: a multi-server model with Poisson arrivals and Exponential service times;
• G/G/s: a multi-server model with General arrival process and General distribution of service times;
• M/M/s/N: a multi-server model with Poisson arrivals, Exponential service times and a finite facility size so that no more than N customers can be present at any time;
• M/M/s Impatient: a multi-server model with Poisson arrivals, Exponential service times and Impatient customers prone to balking or reneging.

The M/M/s model

• In this model arrivals follow a Poisson process, the service times are i.i.d. (independent and identically distributed) and follow an exponential distribution.
• There are s servers (s ≥ 1). In the M/M/s model, there is no balking or reneging, so all arrivals eventually receive service.
• This model is easy to analyze, and software packages usually give exact values for the various performance measures.

Single-Line-Single-Server Model
Queuing models may be formulated on the basis of some fundamental assumptions related to following five features:

• Arrival process
• Queue configuration
• Queue discipline
• Service discipline
• Service facility.

For M/M/1 model following set of assumptions is needed:

• Arrival Process: The arrival is through an infinite population with no control or restriction. Arrivals are random, independent and follow a Poisson distribution. The arrival process is stationary and in a single unit (rather than batches).
• Queue Configuration: The queue length is unrestricted and there is a single queue.
• Queue Discipline: Customers are patient.
• Service Discipline: First-Come-First-Serve (FCFS)
• Service Facility: There is one server, whose service times are distributed as per exponential distribution. Service is continuously provided without any prejudice or breakdown, and all service parameters are state independent.

Two major parameters in waiting line: arrival rate (λ) and service rate (μ).

When arrival rate (λ) is less than service rate (μ), i.e., traffic density (ρ=λ/μ) is less than one we may have a real waiting line situation, because otherwise there would be an infinitely long queue and steady state would never be achieved

• Distribution of arrival is Poisson’s distribution
• Distribution of Service Time follows Exponential law

Following are the lists of parameters

• Mean arrival rate in units per period
• Mean service rate in units per period
• Traffic intensity
• n Number of units in the system
• w Random variate for time spent in the system.

Download Notes for Queuing Model Here