Understanding Waiting Line Models: Single & Multiple Server
Understanding Waiting Line Models
Waiting lines are prevalent in manufacturing and service organizations. Understanding them is crucial for developing solutions to eliminate or minimize their impact.
Waiting lines tend to form when arrival and service patterns are highly variable, creating temporary imbalances between supply and demand.
Structure of Waiting Line Problems
A waiting line problem typically consists of:
- An input, or customer population, that generates potential customers.
- A waiting line of customers.
- The service facility, consisting of a person (or crew), a machine (or group of machines), or both, necessary to perform the service for the customer.
- A priority rule, which selects the next customer to be served by the service facility.
Customer Population
Source of Input
The source of input can be finite or infinite:
- Customers already in line from a finite source reduce the chance of new arrivals.
- Customers already in line from an infinite source do not affect the probability of another arrival.
Customer Behavior
Customers can be patient or impatient:
- Patient customers wait until served.
- Impatient customers either balk (i.e., do not enter the system) or renege (i.e., leave the system before being served).
The Service System
Number of Lines
The system can have a single or multiple lines.
Arrangement of Service Facilities
Common arrangements include:
- Single-channel, single-phase
- Single-channel, multiple-phase
- Multiple-channel, single-phase
- Multiple-channel, multiple-phase
- Mixed arrangement
Priority Rules
The most common priority rule is:
- First-come, first-served (FCFS)—used by most service systems.
Other rules include:
- Earliest due date (EDD)
- Shortest processing time (SPT)
Preemptive discipline allows a higher priority customer to interrupt the service of another customer or be served ahead of another who would have been served first.
Probability Distribution
The sources of variation in waiting-line problems come from the random arrivals of customers and the variation of service times.
Arrival Distribution
Customer arrivals can often be described by the Poisson distribution with mean λT and variance also = λT.
Interarrival times are the average time between arrivals.
Service Time Distribution
Service time distribution can be described by an exponential distribution with mean = 1/μ and variance = (1/μ)2.
Single Server Model
The single server model features a single server, a single line of customers, and only one phase.
Assumptions
- Customer population is infinite and patient.
- Customers arrive according to a Poisson distribution, with a mean arrival rate of λ.
- Service distribution is exponential with a mean service rate of μ.
- Mean service rate exceeds mean arrival rate.
- Customers are served FCFS.
- The length of the waiting line is unlimited.
Single Server Model Formula
Both the average waiting time in the system (W) and the average time spent waiting in line (Wq) are expressed in hours. To convert the results to minutes, simply multiply by 60 minutes/hour. For example, W = 0.20(60) = 12 minutes, and Wq = 0.1714(60) = 10.28 minutes.
Multiple Server Model
The service system has only one phase and multiple channels.
Assumptions (in addition to single-server model)
- There are s identical servers.
- The service distribution for each server is exponential.
- The mean service time is 1/μ.
- sμ should always exceed λ.
Little’s Law
Relates the number of customers in a waiting-line system to the arrival rate and the waiting time of customers.
L = λW or Lq = λWq
Where:
- L = average number of customers in the system
- λ = average arrival rate
- W = average time a customer spends in the system
Service: Estimate W, the average time in facility = W= L customers/(λ customer/hour)
Manufacturing: Estimate the average work-in-process L
Work-in-process = L = λW