Queuing analysis - Lean Six Sigma Resource Management

The dynamic characteristics of many systems have been studied over the past several decades by mathematicians, statisticians, and operations research profession- als. As a result, tools and methods have been developed to analyze specific process workflow applications based on specific assumptions. Also, in many analytical situations the same practical problem can be solved using more than one analyti-

Total Cycle Time Operation 1 + Operation 2 + Operation 3 =

Exponential λ = 90 sec Uniform

α = 10 sec β = 30 sec µ = 60 sec/σ = 10 sec Normal

Normally Distributed Cycle Times(Seconds)

F requency 100 90 80 70 60 50 40 30 120 100 80 60 40 20 0

Histogram of Normally Distributed Cycle Times (Seconds)

Uniformly Distributed CycleTimes (Seconds)

Frequency 30 27 24 21 18 15 12 40 30 20 10 0

Histogram of Uniformly Distributed Cycle Times (Seconds)

Exponential Distributed CycleTimes(Seconds)

Frequency 600 500 400 300 200 100 0 180 160 140 120 100 80 60 40 20 0

Histogram of Exponential Distributed Cycle Times(Seconds)

Total Cycle Time (Seconds)

Frequency 700 600 500 400 300 200 100 160 140 120 100 80 60 40 20 0

Histogram of Total Cycle Time (Seconds)

Minimum

56.87 106.08 Q1 Median 144.37 205.67 Q3 Maximum 729.09

Using Lean Methods to Design for Process Excellence n 147

cal technique. As an example, many workflows can be analyzed using simulation, queuing analysis, or linear programming with similar results, depending on the purpose of the analysis. Although simulation is useful in any analyses, queuing analysis and linear programming are more efficient because they have a firm analytical basis. Figure 5.12 shows an example of the various modeling elements that characterize a queuing system. The queuing system described by Figure 5.12 could represent a process workflow of a bank in which customers arrive, are serviced, and then depart. The arrival pattern of customers into the workflow is specified by their average arrival rate. In addition to the arrival rate, the size of the population from which the people arrive (calling population) may be very large (infinite) or small (finite). Depending on the system modeled, arriving customers may not join the workflow if its waiting line is too long (balking) or, once they join the waiting line, may leave it (reneging). The queuing analysis predicts the average number of people waiting in line, how long they wait on average, the average number of people waiting within the workflow (number waiting in line and being serviced), and how long customers wait on average within the workflow (average time waiting in line and being serviced), as well as the utilization of the workflow’s servers and other relevant statistics.

Table 5.6 shows several common statistics obtained from queuing models. This table shows that much information can be obtained from a very simple study of a system’s queuing characteristics. However, the specific form of the equations providing the information listed in Table 5.6 varies depending on the underlying probability distributions of the queuing system components, which are based on specific characteristics of the workflow modeled. As an example, Table 5.7 shows six major workflow characteristics that help define a queuing model of a process workflow. The distribution of arrivals into the workflow, as well as their behavior

Arriving Population Waiting Line Servers Exit Service Discipline Queue Length

Balk or Leave Queue

table 5.6 Questions Queuing models answer

1. Arrival rate into system (λ) 2. Average units serviced (µ)

3. System utilization factor (λ/µ) (note: λ/µ < 1) 4. Average number of units in system (L) 5. Average number of units in queue (Lq)

6. Average time a unit spends in system (W) 7. Average time a unit waits in queue (Wq)

8. Probability of no units in system (P0)

9. Probability that arriving unit waits for service (Pw)

10. Probability of n units in the system (Pn)

table 5.7 Queuing System Characteristics

Arrival distribution

The arrival distribution is specified by the interarrival time or time between successive units entering the system — also, if the unit balks and leaves the line because it is too long (prior to joining the line) or renegs and leaves the queue because the wait is too long.

Service distribution

The pattern of service is specified by the service time, or time required by one server to service one unit.

System capacity

The maximum number of units allowed in the system; in other words, if the system is at capacity, units are turned away. Service

discipline

There are several rules for a server to provide service to a unit, including first-in-first-out (FIFO), last-in-last-out (LIFO), service in random order (SIRO), prioritization of service (POS), and another general service discipline (GSD).

Channels The number of parallel servers in the system.

Phases The number of subsequent servers in series within a given channel.

Note: Queuing systems are characterized by the arrival distribution of the call-

ing population, the service distribution, the number of services, the number of phases, the service discipline, and system capacity.

Using Lean Methods to Design for Process Excellence n 149

relative to the waiting line, defines the first characteristic of the queuing model. The second characteristic is the distribution of service provided to the arrivals. The third characteristic is the allowable capacity of the system. As an example, some systems cannot accommodate all the arrivals. An example would be a store with a limited amount of parking spaces or a bank drive-up window that allows a finite number of automobiles to wait in line. The fourth characteristic relates to how customers are serviced. Some systems allow first-come-first-served (FCFS) prioritization, whereas others use a different service discipline. The number of channels in the system refers to the total available parallel servers in the model. The number of phases refers to the number of subsequent operations past the first server in a particular channel. Because there are many types of queuing models specified by the characteristics of the system they are designed to model, a concise descriptive notation was developed to make their description easier to understand and interpret by analysts. The clas- sic queuing notation was developed by Kendall and is shown in Table 5.8, along with some modifications to the original notation. Table 5.8 lists some of the more common probability distributions that are used to describe the arrival and service patterns occurring within a workflow. Table 5.9 summarizes our discussion and lists important queuing model characteristics. Once specific quantitative data has been collected relative to the arrival and service rates within the process workflow, as well as its other performance characteristics, a modeling team can build their queuing model. Integral to modeling efforts is building a process map of the workflow according to one of the examples shown in Figure 5.13 or using modified versions of these examples. Once the basic system characteristics have been determined and the process layout has been specified, the team can use off-the-shelf

table 5.8 modified kendall’s notation, (a/b/c): (d/e/f)

Arrival and service distributions

M = exponentially distributed Ek = Erlang type – k distributed D = deterministic or constant G = any other distribution Service

discipline

Note: Kendall’s notation summarizes the modeling characteristics of a queuing

system where a = the arrival distribution or pattern, b = the service distribution or pattern, and c = the number of available servers or channels. Other characteristics can also be added to Kendall’s original notation, such as those by A.M. Lee (d = the service discipline, e = the system’s capacity) and Handy A. Taha (f = size of the calling population, i.e., infinite or finite).

software to quickly analyze the process workflow and obtain the information listed in Table 5.6.

Four common queuing models are shown in Table 5.10. These models are described using simple Kendall’s notation. This is the simplified format that is used to succinctly represent the key characteristics of a queuing model based on the underlying process workflow. There are many other types of queuing models that can be constructed to match your particular workflow. A literature search may be the best way to find a queuing model that matches your team’s specific requirements. The

table 5.9 Summarization of Queuing model Characteristics

Calling Population Service Discipline

1. Deterministic (constant) service

2. Distributed pattern of service

3. Service rules (FIFO, LIFO, SIRO, POS, GSD) 4. Single phase 5. Multiphases 1. Infinite distribution 2. Finite distribution 3. Deterministic arrivals 4. Distributed arrivals

5. Balking or reneging allowed? 6. Single channel

7. Multichannels

Exit

Exit Exit

Single Channel/Single Phase Single Channel/Multi-Phase

Multi-Phase Arriving Population Waiting Line Server Exit Service Discipline Queue Length

Balk or Leave Queue

Multi-Channel/SinglePhase Multi-Channel/Multi-Phase Multi-Phase Arriving Population Waiting Line Server Queue Length

Balk or Leave Queue

Arriving Population Waiting Line Server Server Discipline Server Queue Length

Balk or Leave Queue Arriving Population Waiting Line Server Server Discipline Server Queue Length

Balk or Leave Queue

Using Lean Methods to Design for Process Excellence n 151

(M/M1) model can be used to analyze workflows that are characterized by Poisson arrival and exponential service distributions, a single channel, and first-in-first-out (FIFO) service discipline. An example would be waiting in line at a ticket office where there is one server. The (M/M/k) models can be used to analyze workflows in which there are several parallel servers. Examples would be waiting lines in supermar- kets and banks having several clerks to handle transactions. The (M/G/k) model, if modified with a capacity constraint, can be used to analyze workflows characterized by a finite number allowed in the system. An example would be a Web site designed to handle a limited number of incoming calls. The fourth model, (M/M/1), is classi- cally applied to situations in which the calling population requiring service is small (finite). An example would be a repair shop servicing machines or other equipment.

A simple example of the (M/M/1) queuing model is shown in Table 5.11. The formulas for this model are easy to manually calculate; however, the calculations for other models quickly become complicated and require use of a computer. In this example, customers arrive at the process workflow at an average rate of 20 per hour. It should be noted that this rate fluctuates with some time periods having customers arriving faster or slower than the average rate of 20 per hour. The service rate is 25 customers on average serviced per hour. Again, the average service rate of some time periods will be higher or lower than the average. The variation in arrival and service rates requires that customers periodically wait for service or, alternatively, servers may be idle and wait for customers. A general requirement of all these queuing models is that the average service rate must exceed the average arrival rate.

Arrival and service rates are usually estimated empirically using check sheets or automatically by business activity monitoring (BAM) systems that record customer arrivals and departures from the system. Using this empirical information as well as the other system performance statistics provides useful information relative

table 5.10 four Common Queuing models

Model Type Process Layout Arrival Distribution/Calling Population

Service Distribution/ Service Discipline

(M/M/1) Single channel Poisson arrivals/infinite calling population

Exponential service distribution/FCFS (M/M/k) Multichannel Poisson arrivals/infinite

calling population

Exponential service distribution/FCFS (M/G/k) Multichannel Poisson arrivals/infinite

calling population/ capacity constrained

General service distribution/FCFS (M/M/1) Single channel Poisson arrivals/finite

calling population

Exponential service distribution/FCFS

to workflow performance. This information can be used to minimize customer waiting time and cost, or meet established service levels at minimum cost. As an example, a queuing model can be used in conjunction with marketing research information to build into the workflow a predetermined service level or waiting time to ensure customer satisfaction at the lowest possible operational cost. Once a workflow has been analyzed in this manner, many options to improve it will usually become apparent to the team.

Another example that shows the advantages of using queuing analysis to model a dynamic workflow is shown in Table 5.12. Although there are many books rec- ommending that a workflow should have excess capacity, the reason is not always clearly shown. However, Table 5.12 shows why excess capacity may be useful for some systems. As the utilization of the machine increases, the average waiting time for a unit to be serviced by the machine increases rapidly. However, designing a process workflow with low utilization may not be a good alternative. An alternative strategy would be to use low-cost parallel machines that are activated rather than utilized as demand fluctuates instead of utilizing one very large and expensive machine that is inflexible to demand fluctuations. Another example would be using information technology to transfer incoming customer calls to call centers that have excess capacity rather than to a location at its maximum capacity level.

table 5.11 example of Simple Queuing model (m/m/1)

Customers arrive at a repair shop at an average rate of λ = 20 per hour; the average service rate is 25 customers per hour. Assume Poisson arrival distribution/exponential service distribution/a single channel/FCFS service discipline/no maximum on number of systems/infinite calling population.

1. Arrival rate into system (λ) λ = 20 per hour 2. Average units serviced (µ) µ = 25 per hour 3. System utilization factor (λ/µ) (note: λ/µ < 1) λ/µ = 0.80 = 80% 4. Average number of units in system (L) L = Lq + (λ/µ) = 4.0 units

5. Average number of units in queue (Lq) Lq = λ2/[µ(µ – λ)] = 3.2 units

6. Average time a unit spends in system (W) W = Wq + (1/µ) = 0.20 hours

7. Average time a unit waits in queue (Wq) Wq = Lq/λ = 0.16 hours

8. Probability of no units in system (P0) P0 = 1 – (λ/µ) = 0.20 = 20%

9. Probability that arriving unit waits for service (Pw)

Pw = λ/µ = 0.80 = 80%

Using Lean Methods to Design for Process Excellence n 153

In document Lean Six Sigma Resource Management (Page 169-176)