Our organizational and socioeconomic systems often require that we “wait” for a service or a resource. This waiting process is achieved through the process of queuing. Bank tellers, gasoline service stations, post offices, grocery check-out lanes, ATM machines, food service counters, and benefits offices are common examples of where we experience queues. Not all queues involve humans. In many cases, the objects and subjects in a queue are themselves service elements. Thus, we can have an automated service waiting on another automated service. Electronic communication systems often consist of this type of queuing relationship. There can also be different mixtures or combinations of humans and automated services in a composite queuing system. In each case, the desire of industrial engineers is to make the queue more efficient. This is accomplished through computational analyses. Because the goal of IE is centered on improvement, if we improve the queuing process, we can make significant contributions to both organizational and technical processes.
In terms of a definition, a queuing system consists of one or more servers that provide some sort of service to arriving customers. Customers who arrive to find all servers busy generally enter one or more lines in front of the servers. This process of arrival–service–
departure constitutes thequeuing system. Each queuing system is characterized by three components:
• The arrival process
• The service mechanism
• The queue discipline
Arrivals into the queuing system may originate from one or several sources referred to as thecalling population. The calling population (source of inputs) can be limited or unlimited. The arrival process consists of describing how customers arrive to the system.
This requires knowing the time in-between the arrivals of successive customers. From the inter-arrival times, the arrival frequency can be computed. So, if the average inter-arrival time is denoted asλ, then the arrival frequency is 1/λ.
Theservice mechanismof a queuing system is specified by the number of servers with each server having its/his/her own queue or a common queue. Servers are usually denoted by the letters. Also specified is the probability distribution of customer service times. If we letSidenote the service time of theith customer, then the mean service time of a customer can be denoted byE(S). Thus, the service rate of a server is denoted asμ=1/E(S). The variables,λandμ, are the key elements of queuing equations.
Queue disciplineof a queuing system refers to the rule that governs how a server chooses the next customer from the queue (if any) when the server completes the service of the current customer. Commonly used queue disciplines are:
• FIFO: first-in, first-out basis
• LIFO: last-in, first-out
• SIRO: service in random order
• GD: general queue discipline
• Priority: customers are served in order of their importance on the basis of their service requirements or some other set of criteria.
Banks are notorious for tweaking their queuing systems, service mechanisms, and queue discipline based on prevailing needs and desired performance outcomes. The per- formance of a queue can be assessed based on severalperformance measures. Some of the common elements of queue performance measures are:
• Waiting time in the system for theith customer,Wi
• Delay in the queue for theith customer,Di
• Number of customers in the queue,Q(t)
• The number of customers in the queue system at timet,L(t)
• The probability that a delay will occur
• The probability that the total delay will be greater than some predetermined value
• The probability that all service facilities will be idle
• The expected idle time of the total facility
• The probability of balking (un-served departures) due to insufficient waiting accommodation
Note that
Wi=Di+Si
L(t)=Q(t)+number of customers being served att.
If a queue is not chaotic, it is expected to reach steady-state conditions that are expressed as
• Steady-state average delay,d
• Steady-state average waiting time in the system,w
• Steady-state time average number in queue,Q
• Steady-state time average number in the system,L d= lim
n→∞
n
i=1Di
n , w= lim
n→∞
n
i=1Wi
n , Q= lim
T→∞
1 T
T
0
Q(t) dt, L= lim
T→∞
1 T
T
0
L(t) dt,
Q=λd, L=λw, w=d+E(S).
The following short-hand representation of a queue embeds the inherent notation associated with the particular queue:
[A/B/s] :{d/e/f}
whereAis the probability distribution of the arrivals,Bthe probability distribution of the departures,sthe number of servers or channels,dthe capacity of the queue(s),ethe size of the calling population, andf the queue ranking or ordering rule.
A summary of the standard notation for queuing short-hand representation is pre- sented below:
M random arrival, service rate, or departure distribution that is a Poisson process E Erlang distribution
G general distribution
GI general independent distribution
D deterministic service rate (constant rate of serving customers) Using the above, we have the following example:
[M/M/1] :{∞/∞/FCFS},
which represents a queue system where the arrivals and departures follow a Poisson distribution, with a single server, infinite queue length, infinite calling population, and the queue discipline of First Come, First Served (FCFS). This is the simplest queue system that can be studied mathematically. It is often simply referred to as theM/M/1 queue.
The following notation and equations apply:
λ arrival rate μ service rate
ρ system utilization factor (traffic intensity)=fraction of time that servers are busy (=λ/μ)
s number of servers
M random arrival and random service rate (Poisson) D deterministic service rate
L the average number of customers in the queuing system Lq the average queue length (customers waiting in line)
Ls average number of customers in service W average time a customer spends in the system Wq average time a customer spends waiting in line
Ws average time a customer spends in service
Pn probability that exactlyncustomers are in the system
For any queuing system in steady-state condition, the following relationships hold:
L=λW Lq=λWq
Ls=λWs
The common cause of not achieving a steady state in a queuing system is where the arrival rate is at least as large as the maximum rate at which customers can be served.
For steady state to exist, we must haveρ<1.
λ μ=ρ,
L=(1−ρ) ρ (1−ρ)2
= ρ 1−ρ
= λ μ−λ, Lq=L−Ls
= ρ 1−ρ−ρ
= ρ2 1−ρ
= λ2 μ(μ−λ).
If the mean service time is constant for every customer (i.e., 1/μ), then W=Wq+ 1
μ
The queuing formulaL=λW is very general and can be applied to many situations that do not, on the surface, appear to be queuing problems. For example, it can be applied to inventory and stocking problems, whereby an average number of units is present in the system (L), new units arrive into the system at a certain rate (λ), and the units spend an average time in the system (W). For example, consider how a fast food restaurant stocks, depletes, and replenishes hamburger meat patties in the course of a week.
M/D/1 case (random arrival, deterministic service, and one service channel) Expected average queue length,Lq=(2ρ−ρ2)/2(1−ρ)
Expected average total time in the system,W=(2−ρ)/2μ(1−ρ) Expected average waiting time,Wq=ρ/2μ(1−ρ)
M/M/1 case (random arrival, random service, and one service channel) The probability of having zero vehicles in the systems,P0=1−ρ The probability of havingnvehicles in the systems,Pn=ρnP0
Expected average queue length,Lq=ρ/(1−ρ) Expected average total time in system,W=ρ/λ(1−ρ) Expected average waiting time,Wq=W−1/μ
It can be seen from the above expressions that different values of parameters of the queuing system will generally have different consequences. For example, limiting the queue length could result in the following:
• Average idle time might increase;
• Average queue length will decrease;
• Average waiting time will decrease;
• A portion of the customers will be lost.