Chapter 5

More Complex Multiple-Channel Models

In many service businesses and manufacturing line applications, the basic M/M/s model, like the M/M/1 model discussed in chapter 4, does not always provide useful results that appear to agree with reality. In addition to the reasons listed at the beginning of chapter 4, there are added factors as a result of using more than one server:

• More complex formulas to deal with, particularly for situations with limited queues or limited calling populations
• The opportunity to separate customers into different classes
• The opportunity to treat customers with different priorities
• The effects of using different line configurations
• Server capability differences

Equations 3.1 through 3.5 for the basic M/M/s model are restated here in a different form using ρ instead of λ/µ.¹ This allows for easier computation of basic performance values.

and

The time spent waiting in line and the total time in the system can be easily determined for a given average arrival rate using Little’s Law after you have calculated the number of customers in line and in the system. The data tables in appendix C provide the values of P₀, L_q, and L for various combinations of ρ values and number of servers up to 10. You are encouraged to set up your own spreadsheet to do these calculations for values of r not listed and/or larger numbers of servers.

Looking at the values listed in appendix C, one can observe that as r approaches a value equal to the number of servers chosen, the length of the queue rapidly increases. The implication is, like the situation for a single-channel model, to not plan for an overall utilization factor, ρ/s, greater than 90% if you want to have some reserve capacity for unexpected increases in the average customer arrival rate. Also, the tables do not list values for every combination of r and number of servers because I have arbitrarily limited the values to what I consider to be practical solutions. For example, having more than 3 servers for r < 1 would be a waste of resources in most situations.

Looking at the values of L, some of my business students ask why one would want to plan for fewer customers in the system than the number of servers available. I remind them that the L values are a steady-state average over time, but there will often be clusters of customers arriving within a limited period in a real-life situation. Without some reserve capacity, the customer wait at those times can be unacceptably long. The managerial view here is to have servers whose skills are flexible enough so that they can do other things for the business when the number of customers is sparse. Businesses that are good at this are grocery stores, banks, and the post office—either calling staff to the checkout lines when business is heavy or training employees to open another service window when a customer line reaches a specified length.

Limited Capacity

This situation using multiple-channel models is often applied to call centers where the number of people who can be put on hold awaiting the next available representative is limited. When that capacity is filled with waiting customers, new customers encounter a busy signal and in effect are blocked from entering the system. Estimating how many customers are being turned away is necessary to determine how many representatives (and phone lines) are needed to provide a desired level of service (the percentage of customers who do not encounter busy signals; that is, those not blocked from entering the system).

When the number of customers is equal to the number of servers, there are no customers waiting in line. This condition occurred in the early days of queuing theory when the most common application was telephone exchanges using operators to make connections for callers. The technology for allowing customers to wait in an electronic line was not yet developed, and callers got a busy signal if the operators were already engaged with helping previous callers. Such queuing models were called an Erlang loss system because potential customers were lost when all the servers were busy. When a waiting line application allowed customers waiting for service to form a line, it was then called an Erlang delay system because those customers were not lost when all the servers were busy; they were just delayed in receiving service.

The set of average performance equations for the M/M/s/K model with a system capacity of K customers is as follows (K must be ≥ s):

• Utilization factors (note that ρ can be greater than 1 if the number of servers is greater than 1):

  ρ_s = λ/sµρ = λ/µ

• Probability of zero customers in the system:

  (5.6)

• Probability of exactly n customers in the system:

  (5.7)

• Probability of customers in the system greater than the number of servers (probability that the servers are busy):

  

• Probability that a customer will be turned away: P_K = P₀ρ^K/s!s^K-s
• Average number of customers waiting in the queue (not yet being served):

  (5.8)

  There is no solution in Equation 5.8 when ρ_s = ρ/s = 1. A solution can be found using L’Hospital’s Rule² twice to obtain the limit for L_q when ρ/s approaches the value of 1. Although the resulting numerator terms can be complex, the denominator term reduces to 2s!s for ρ/s = 1.

• Average number of customers in the system:

  (5.9)

• Effective arrival rate: λ′ = λ(1 – P_K)
• Average total time customers spend in the system: W = L/λ′ = L/(λ(1 – P_K))
• Average time customers wait in the queue before being served:

  W_q = L_q/λ′ = L_q/(λ(1 – P_K))

Like the discussion in chapter 4 for the M/M/1 model, we need to use an effective arrival rate (λ′) for Little’s Law. This is the arrival rate minus the customers or items turned away by the limit in line capacity. The assumption is that those customers are lost and do not return. Because we lose some customers, we can allow the average arrival rate to be as high as the average service rate, which explains the conditional equations for P_n expressed by Equation 5.7. P_K gives us a value for lost business that can be used to compare with the cost of adding additional line capacity. P_K is sometimes called the blocking probability.

Limited Calling Population

Example 4.3 described a repair service with one person responsible for maintaining 10 items. When the number of customers or items becomes significantly larger, as illustrated in that example, more than one person is usually needed to provide an adequate turnaround time for items needing repair or maintenance. There are two approaches for analyzing the performance of adding staff:

1. If an item requiring service is of the type where a service crew can be effectively used (such as maintenance on an airplane or a truck), then the repair crew is treated as a single server where adding a member to the crew hopefully reduces the average service time per item. In this case, the M/M/1/∞/N model discussed in chapter 4 would still apply, but we would use a shorter repair time associated with the increased crew staff for our calculations.
2. However, in many repair and maintenance situations, the item requiring service can be effectively worked on by only one person. Hence, we need to use the M/M/s/∞/N model to determine the average performance expected if we decide to add one or more servers to improve performance. The disadvantage is that the equations become much more challenging to use, and additional care is required to avoid mathematical errors.

Recalling the use of an unit arrival rate and associated unit utilization factor as discussed for the single-channel limited population model in chapter 4, the set of average performance equations for the M/M/s/∞/N model with a limited population of N customers or items to be served is as follows (N must be ≥ s):

• Arrival rate per unit in the population: λ_u
• Unit utilization factor: ρ_u = λ_u/µ
• Probability of zero customers in the system:

  (5.10)

• Probability of exactly n customers in the system:

  

• Probability of customers in the system equal to or greater than the number of servers (probability that the servers are busy):

  

• Average number of customers waiting in the queue (not yet being served):

  (5.12)

• Average number of customers in the system:

  (5.13)

• Effective arrival rate: λ′ = λ_u (N – L)
• Average total time customers spend in the system: W = L/λ′ = L/(λ_u (N – L))
• Average time customers wait in the queue before being served:

  W_q = L_q/λ′ = L_q/(λ_u(N – L))

• Probability of no waiting time for the next arrival:

  

Before returning to Example 4.3 to see what improvement is possible for a greater number of printers by adding staff, let us consider how we might develop our own finite queuing tables³ using Equations 5.10 through 5.13 in a spreadsheet program like Excel. Like the multiple-server table in appendix C, we can obtain some useful values of P₀, L_q, and L for various combinations of N, ρ, and s. Like the finite queuing tables published by Peck and Hazelwood (1958), it will be best if we prepare separate tables for different calling population sizes. But unlike Peck and Hazelwood’s tables, we can set up our Excel solutions so that we can obtain P₀, L_q, and L directly rather than going through the use of some intermediary variables.

Unlike the equations provided in chapter 3 for the basic M/M/s model, the equations for P₀, L_q, and L become too complicated to formulate in a single Excel cell when the population is limited because of the extensive summations required in the formulas. Therefore, if you want to set up your own set of tables, you will need to use some additional columns to compute the summation values for specific combinations of ρ_u and s for your specific population N. Recall that we did this in chapter 4 when we analyzed a single-channel model with a limited line capacity K (see Figure 4.3).

An example spreadsheet solution is illustrated in appendix C for Example 5.1, where we return to the Campus Reboot situation discussed in chapter 4 and consider adding one or more servers to that repair service.

To obtain a full analysis for a more informed recommendation, we need to consider the trade-offs in the costs of hiring another repair person versus having a faculty member wait longer for a repair to be completed.

Different Classes of Customers

An important consideration in improving service performance when working with multiple-channel waiting lines is recognizing whether or not you have different classes of customers with different service distributions. When more than one server is required to provide acceptable service, an added decision is whether you have each server handle any customer that arrives or designate servers to specific classes of customers. A common example is the use of express and regular checkout lines at a grocery store when the store is large enough to justify having more than one checkout clerk.

More detailed aspects about taking advantage of customer classes for improving service performance is discussed in chapter 6, where we also can take into account other managerial considerations.

Different Customer Priorities

Multiple-channel waiting lines provide additional options for handling customers with different service priorities. When there is only one server, priorities can be handled only by manipulating the order of customers in the queue. This is a frequent concern in a small hospital emergency room with only one doctor on duty, where the most gravely ill patient must be treated first. Obviously, when there is more than one of these patients at a time, then tough choices have to be made as to which is treated first. That decision is clearly beyond the scope of the level of analysis discussed here.

When more than one server is available, several options are available for managing priority needs. In one aspect, priority defines a class of customers that are handled in the same way as we decide to handle other classes. These approaches will be discussed in more detail in chapter 6—where we will discuss class management.

Different Line Configurations

More servers also allow some creativity in how the line(s) of customers are arranged, particularly when some servers are dedicated to a particular customer class or priority. In addition to improving service performance, important considerations for line configuration design are the psychological effects on customers and opportunities to inform customers as to what they can do to help speed up the service process.

A common example of different line configuration approaches used in many banks, post offices, and airport security check-ins was shown in Figure 3.1. From a strict average performance business viewpoint, both the separate line per server and one line feeding all servers (the “snake”) configurations have the same average values for service performance.

The difference between the two configurations is found in individual customer waiting time experiences and psychological impressions. Because of this, the configuration choice is more dependent on managerial attitudes and prior experience rather than actual numbers. We will discuss this topic in more detail in chapter 6 where we will focus on managerial concerns.

Service Capability Variations

In chapter 3 and this chapter, our analysis has assumed a consistent service rate for each server. In reality, we all know that this assumption is often invalid based on our experience in a line where a new teller or checkout clerk is learning on the job. Most service activities are repetitive in nature, and repetition makes a person more proficient (faster and more accurate) as he or she gains experience doing that task. Such improvement is more dramatic at first and slowly diminishes in magnitude over time until the person’s performance essentially flattens out to a steady level. Some good examples are learning how to ride a bicycle or memorizing commonly used reference values or coffee recipes instead of having to look them up each time you need them.

Learning curves can be used to estimate the time to reach a given proficiency if there is information available as to what percentage of improvement over time is typical for a given job content. In general, the more repetition and less variety of tasks involved, the faster the improvement rate. The learning curve equation is based on the concept that every time the total number of repetitions of a task is doubled, the amount of time per repetition is reduced by a constant percentage. The equation for a learning curve takes the following form:

Here T_n is the time to do the nth repetition, T₁ is the time to do the first repetition, and the % sign is the learning percentage value. A learning percentage value of 80% (0.8) corresponds to a 20% reduction in time per repetition every time the total number of repetitions doubles. So a higher learning percentage corresponds to a slower improvement in proficiency (100% = no improvement with repetitions).

If you have some data regarding how long each successive repetition takes for your situation, you can estimate your typical learning percentage in two different ways. The first method is more universal but a bit more complicated:

The other method is to take the ratio of any pair of repetition times, where the number of repetitions for one time is double or half the number of repetitions for the other time. Then the learning percentage is the shorter time divided by the longer time. That is, the % = T_x/T_x/2 or T_2x/T_x.

Figure 5.1 shows several learning curves with different learning percentages. A table of the values used for the different learning curve percentages and numbers of repetitions in Figure 5.1 is in appendix C.

Is 3 months too long for a service employee to become proficient? Shortening the amount of time required would be to the advantage of any business. It obviously depends on the skills needed and the complexity of the tasks to be done. An earlier edition of a popular college operations management textbook⁴ used a bakery to illustrate the use of various manufacturing process strategies in a single business. In the video of the bakery operations that accompanied the text, the bakery manager commented on the typical training time for new employees for each process. The highly automated bread process required only a few weeks, work in a pastry shop required about 3 months, and a cake decorator required about 6 months to become proficient. Because effective training approaches often are dependent on the right managerial support, we will reserve further discussion about such methods for chapter 6.