Waiting lines are an important consideration in capacity planning.
Waiting lines tie up additional resources (waiting space, time, etc. ); they decrease the level of customer service: and they require additional capacity to reduce them. 2. Waiting lines occur whenever demand for service exceeds capacity (supply).
Even in systems that are underloaded, waiting lines tend to form if arrival and service patterns are highly variable because the variability creates temporary imbalances of supply and demand.3. All of the waiting line models presented in the chapter (except the constant service time model) assume, or require, that the arrival rate can be described by a Poisson distribution and that the service time can be described by a negative exponential distribution. Equivalently, we can say that the arrival and service rates must be Poisson, and the interarrival time and the service time must be exponential.
In practice, one would check for this using a statistical Chi Square test: for problems provided here and in the textbook, assume that these distributions hold.Note that if these assumptions are not met, alternate approaches (e. . , intuition, simulation, other models) should be considered.
4. Much can be learned about the behavior of waiting lines by modeling them. A wide variety of models are presented in the text, different models pertain to different system characteristics. 5.
A major distinction in waiting line models relates to whether the number of potential arrivals to the system is limited (finite) or unlimited (infinite). Perhaps the classic example of a finite source system is the machine, repairperson problem, wherein the server or servers handle calls for repairs on a small, fixed number of machines.Note that the definition of terms in Table 18-6 in the text follows this somewhat (e. g.
, average number running). Other examples of finite source systems include passengers on a plane who might request assistance or information from a steward or stewardess, a sales rep who handles a small set of customers, and a telephone operator who places outgoing calls for residents of a small hotel. Examples of infinite-source systems are plentiful: service stations, banks, post offices, restaurants, theaters, supermarkets, libraries, stop signs, and telephone switchboards.6.
Once a situation has been modeled and the relevant data gathered (e. . , an-arrival rate, service time, number of servers, etc. ), formulas and/or tables can be used to obtain information about the system such as the expected number waiting for service, the expected waiting time, the maximum line length, system utilization, and so on.
This information can be used to compare various system alternatives (e. g. , one server versus two servers, different equipment possibilities, and so on) with respect to cost and impact on waiting times, etc. 7.
The goal in queuing analysis is to develop a system in which the sum of capacity costs and waiting costs is minimized.Sometimes this goal may be taken in an absolute sense or the goal may be to minimize total costs given a minimum level of customer service specified by management. 8. Most of the models described in the chapter assume arrivals are processed on a first-come, first-served basis (FCFS).
FCFS is one example of queue discipline, or the order in which customers receive service. Sometimes, however, customers are processed on a priority basis rather than FCFS. In a hospital emergency room, for example, seriously ill or injured persons are attended to while less seriously ill persons wait.Although arrival rates can vary by priority class, the model treats service rates as being the same for all priority classes.
9. A very important concept for the analysis of waiting lines is Little’s law, which states that in a stable system, the average number of customers in line or in the system is equal to the average customer arrival rate multiplied by the average time in line or in the system. That is, Ls = (Ws, and Lq = (Wq. The relationships are independent of any probability distribution and require no assumptions about when customers arrive or are serviced, or the order in which they are served.
It also means that knowledge of any two of the three variables can be used to obtain the third variable. For example, knowing the arrival rate and the average number in line, one can solve for the average waiting time. STUDY TIPS Much of the material is highly mathematical. Do not to let this overwhelm you.
Note that many of the problems can be solved using tables rather than formulas, and that many of the formulas do not involve extensive computation. The chapter presents two kinds of queuing models, one for infinite source (unrestricted entry) systems, and another for finite source systems.There are four infinite source models and one finite source model. Instead of trying to absorb the entire chapter at once (or even a major portion of it), begin with the general discussion of waiting lines, and then focus on a few infinite source models (e.
g. , single channel and single channel with constant service, which represent the first two infinite source models), leaving other models to another time. This can be reinforced by attempting some problems at the end of the chapter which relate to the models you have focused on.Moreover, if you have done a good job in studying a particular model, you should be able to ascertain whether or not a given problem can be solved using the model.
TIPS FOR SOLVING PROBLEMS 1. A key element of infinite source models is the number of channels. Generally, there is one channel for every server. The exception would be if the servers work in teams; then there is one channel for every team.
2. One key measure for infinite source problems is the arrival rate, which is symbolized by Greek letter [pic](lambda). The formulas and tables require rates, although times may be given, necessitating a conversion to a rate.Moreover, arrival and service rates must be in the same units (e.
g. , both in customers per hour, both in customers per minute). Consider these examples for arrivals: a. 10 customers per hour arrive at the facility.
This is okay, it is a rate and [pic]= 10. b. Customers arrive at an average of one every 20 seconds. This gives the time between arrivals, which is symbolized by [pic].
To find the arrival rate, use the information to determine the number of customers, say, per minute. Thus, | |60 seconds/minute |= 3 customers per minute =[pic]. | | |20 seconds | | c. The interarrival time is 90 seconds.
Again, we have the time etween arrivals. Dividing this into 60 seconds per minute, we get . 667 customers per minute. Multiplying this by 60 minutes will give us the rate per hour: .
667/minute x 60 minutes/hour = 40 customers per hour. 3. The other key measure for infinite source problems is the service rate, which is symbolized by Greek letter [pic] (mu). Again, the formulas and tables require rate, although times may be given, necessitating a conversion to a rate.
Arrival and service rates must be in the same units (e. g. both in customers per hour or both in customers per minute).a.
Three customers per hour get service. This is okay, it is a rate and [pic]. . Service time is 15 minutes.
Service times are symbolized by [pic]. This means that four customers can be served in an hour: | | 60 minutes/hour |= 4 customers per hour = [pic]. | | |15 minutes/customer | | c. The service time is 12 minutes.
The service rate is 1/12 per minute. Multiplying by 60 gives us the hourly rate: 60 (1/12) = 5 customers/hour. 4. When a problem refers to the average number waiting in line, check to see if it pertains to all arrivals, or just those that actually wait, since customers that arrive when the service facility is idle would not have to wait, thereby lowering the average waiting time of all arrivals.
Use the formula or table for [pic] if the average for all customers who arrive is required; and use [pic] if the average waiting time only for those that actually wait is required. Note that in a heavily loaded system, most customers will have to wait, in which case the two averages, [pic] and [pic] will be approximately equal. More generally, [pic] will be greater than [pic] since [pic] includes customers who do not have to wait.
