Modeling Processes

Modeling processes enables experimentation and evaluation of alternative process designs under varying process conditions. The ten steps shown in Table 5.4 help implement process modeling. The first step is to bring together a group of people trained to build and analyze process simulations. They should have skills in engineering or statistics. Training provided by consultants or suppliers may also be useful. The second step is to create a list of use cases or examples of where the model can be applied to create benefits such as reductions in lead time or costs and process simplification. There are modeling methodologies that can be used depending on the type of process and questions that need to be answered in the analysis. Examples include simulation, queuing analysis, linear programming, and customized models and algorithms for specific applications.

Once the team has selected the type of modeling methodology, it selects the software and hardware. Modeling software has been developed for a wide range of applications based on differing assumptions. These assumptions range from simple to complex. It is always more efficient to use


Ten Steps of Process Modeling


1. Organize a group of people who have been trained to build simulation models.

2. Develop a list of areas in which the model’s methodology can be realistically focused.

3. Research and select off-the-shelf modeling software and associated hardware to match expected process applications (e.g., manufacturing, service systems, warehousing, logistics, etc.).

4. Develop a library of use cases and applications that can be used as examples of applying the model within your process.

5. Develop the underlying model structure including its goals and objectives, system constraints, and parameter settings.

6. Determine probability distributions of the metrics and time span of the model.

7. Develop decision rules, including initial and final states of the model.

8. Develop plans to obtain the necessary process data to test model accuracy.

9. Analyze the output of the model using statistical tests to determine the significance of the model’s output.

10. Document and communicate the model’s results and develop plans to implement solutions as practical.

software designed specifically for your application because it will already have relevant models that will be easier to modify and interpret. As an example, there may be specially designed modeling software options for manufacturing, financial services, call centers, warehousing, inventory management, distribution networks, and others. It is easier to use software that has easily configurable system elements and reflects your process rather than having to create new models. It is also a good idea to create a library of use cases and examples. This helps communicate the advantages of process modeling to stakeholder groups by showing them alternate process designs

The team next builds its process model. This involves documenting the process from start to finish, including its serial and parallel operations, decision points, rework loops, constraints, parameter levels, and metrics. In addition, probability distributions are determined for the model’s inputs and outputs (i.e., the metrics). The model’s structure, time span, and decision rules should mirror the important characteristics of the actual process. After the model is created and run, the process data are collected to test its accuracy. Finally, the analyses are documented in an appropriate format and communicated to stakeholders with recommendations and practical solutions for process optimization and improvement.

Figure 5.5 shows that process models are virtual representations that correspond to a real process. The interrelationships of the operations within a process may initially be unknown or poorly understood. The dynamic or complex performance of integrated processes may not be completely understood without creating models. Operational relationships are usually complicated and not obvious because real systems have ambiguity and time delays between the event occurrences and when their outputs are seen. This makes it difficult to understand relationships between cause and effect within a process. Therefore, mapping of processes along with their parameters and decision rules into a virtual model will be useful for understanding and improving operational performance. The advantages of a using a process model are that its structure and operational components can be easily modified and event frequency can be compressed to enable numerous evaluations of process modifications to identify an optimized final state.

Simulation is useful for modeling diverse types of processes. Within a process, system capacity can be evaluated under various constraints, comparing system performance between several alternative designs and


Process complexity.

conducting sensitivity analyses to determine the impact of varying one or more key process input variables (KPIVs) on key process output variables (KPOVs). A simulation model can be flexibly designed to evaluate event probabilities based on their underlying probability distributions. Or, if the goal is maximization or minimization of an objective function (i.e., a KPOV) and the model has clearly defined constraints, then linear programming might be useful for modeling the process. Queuing models provide additional tools and methods to model processes if they satisfy certain assumptions and fit predefined criteria.

The first step for creating a simulation model is asking what the expectations from the analysis are. Other relevant considerations include the project’s budget and schedule. Developing simulation models are interactively modified to evaluate performance. The second step is defining the scope of the simulation relative to the operations being modeled. In other words, where does the model begin and end? Then the underlying functional relationships (i.e., Y =f(X), where У is the KPOV and X is the KPIV) are evaluated. These form the basis of the model. The relationships between the model KPOVs and the KPIVs must be defined in terms of these Y = f(X) transfer functions as well as applicable decision rules and constraints. The fourth step is the collection of process data to help structure and analyze the model. The sources include process maps, historical process data related to throughput rates, yields including rework and scrap, machine and direct labor cycle times, downtimes, lot sizes, inventory levels, floor layouts, and other relevant operational data, depending on the process. The specific data collected and analyzed should correspond to the questions that need to be answered by the analysis.

It will be useful to discuss the simple example shown in Figure 5.6 to demonstrate the basic steps for conducting a simulation. The first step is



defining the functional relationships between the output and its inputs as represented by the expression Y =f{X). In step two, the system’s clock is set to time t = 1, an event is simulated, and the model’s statistics are updated by the model’s algorithm, which represents the business rules. If the simulation is not at its terminal time, the clock is advanced to the next time period (f = 2) and the simulation cycle continues until the terminal time. Statistics are collected based on the functional form of the model, which depends on the probability of the event occurrence at each operation.

We will discuss two simple examples, a single operation and a work- flow that consists of three sequential operations. In Figure 5.7, a single operation and independent variable, cycle time, has been assigned probability values over its observed range of cycle times. A specific cycle time of twelve days is used as an example. This simulation model is based on a uniform distribution and generates random numbers with a uniform occurrence having a probability between 0 and 1. These random numbers are transformed using the cumulative density function (cdf) of cycle times of the actual observed distribution. The cdf has a range between 0 and 1 based on the original probability density function (pdf). The relationship of a cdf to an independent variable can be discrete or continuous depending on the pdf, which is based on the distribution assumption. This example uses a uniform distribution. The functional relationship between cycle time and its occurrence probability has been discretely defined in Table 5.5. A random number in the range of greater than 0.539828 and less than

0.617911 is defined as a discrete cycle time of 12 days. Using a continuous “cdf,” we also map a one-to-one relationship between the continuous random variable in the range between 0 and 1 to a specific cycle time. This is shown in Figure 5.7.

Figures 5.8, 5.9, and 5.10 show three common probability distributions that are used in simulations. There are others for specific applications that have different distribution assumptions. Once the statistical sampling shows the pattern or distribution of the variable to metric being analyzed, which in this example is cycle time, the empirical data are fit to a standard probability distribution using goodness-of-fit testing methods. Once a match to a specific probability distribution is found, the formula for the calculating events is determined and built into the model at the correct operation. Recall this formula is calculated using the cdf of the pdf.


How simulation works. Every event has a probability. If we know its probability density function (pdf) then we can calculate the cumulative density function (cdf). When we simulate a random event between 0 and 1, we can use the cdf to generate an event from the original pdf. If the random number is less than 0.617911 and higher than 0.539828, then the event is 12 days.


How Simulation Works

If the random number is less than 0.617911 and higher than 0.539828 then the event is “12”

Value ofX


Cumulative Probability




























Figure 5.11 shows how these concepts are applied in practice. In this example, the cumulative cycle time through the process is calculated by adding the simulated cycle times at each sequential operation. Operation 1 is uniformly distributed, with a lower cycle time of 10 seconds and an upper cycle time of 30 seconds. These two parameters specify a unique uniform distribution for Operation 1. Operation 2 is a normal distribution, with a mean cycle time of 60 seconds and a standard deviation of 10 seconds. Operation 3 is exponentially distributed with a mean cycle time of 90 seconds. The cumulative cycle time, across the process, is the sum of the three operational cycle times. The analysis shows the median cycle time through the process is -144 seconds with a right-skewed distribution of cycle times. This simple example shows the underlying logic behind a simulation model. This method, however, would be tedious to apply to more complicated processes with several operations, parallel paths, decision points, and rework loops. Off-the-shelf simulation software can be used to simplify and reconfigure the model easily, depending on the objectives of the analysis.

The dynamic performance of certain systems has been studied by mathematicians, statisticians, and operations research professionals. These


Uniform distribution. Assume the uniform distribution had a maximum of 6 and a minimum of 1. If a random number of 0.5 was generated by the computer, the value of X, would be 3.5, which is the mean of this distribution.

studies created analytical models called queuing or waiting line models. If certain assumptions can be met, queuing models are useful and easy to apply. The same type of model can often be used in different processes that have common assumptions. Also, in many analytical situations, the same practical problem can be solved using more than one analytical technique. As an example, some processes can be analyzed using simulation, queuing analysis, or linear programming with similar results. Although simulation is useful in almost any analysis, queuing analysis and linear programming are usually more efficient if they fit because they have an an exact analytical solution.

Figure 5.12 shows an example of the various components that characterize a queuing model. This queuing model could be used to describe a process for a bank, a restaurant, or any system in which customers arrive, are serviced, and depart. The specific queuing model depends on


Normal distribution.


Exponential distribution.


Estimating lead time across several operations. Simulation is especially useful when the output at every step is not normally distributed and thus cannot be analytically combined into a total. Ql = first quartile; Q3 = third quartile.

assumptions. These include the arrival distribution of customers into the process as specified by an average arrival rate. A second assumption is the size of the incoming arrival population (calling population). It may be very large (infinite) or small (finite). The analysis varies depending on the answer. Arriving customers may choose not to join the line if the waiting line is too long (i.e., balking), or, once they join the waiting line, they may leave it (i.e., reneging).

A queuing model calculates several statistics, including the average number of people waiting in line, the average wait time, the average number of people waiting within the whole process (i.e., the number waiting in line and the number being serviced), how long customers wait on average


Common elements of a queuing system.

within the process (i.e., average time waiting in line and being serviced), and the utilization of the processes’ servers and other relevant statistics.

Table 5.6 lists common statistics obtained from queuing models. Much information can be gained immediately once the process assumptions are known, but the specific form of the equations providing the information listed in Table 5.6 vary based on the underlying probability distributions, which depend on specific assumptions or characteristics


Questions Answered by Queuing Models


1. Arrival rate into system {X)

2. Average units serviced (ц)

3. System utilization factor (А./ц) note: Xlx < 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 (P„)

9. Probability arriving unit waits for service (Pw)

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


Queuing System Characteristics



Arrival distribution:

'Ihe arrival distribution is specified by the inter-arrival time or time between successive units entering the system; this is also affected if the unit balks and leaves the line because it is too long (i.e., prior to joining the line) or reneges and leaves the queue because the wait is too long (i.e., after joining the line).

Service distribution:

'I he pattern of service is specified by the service time or time required by one server to service one unit.

System capacity:

'I he maximum number of units allowed in the system. In other words, if the system is at capacity, units are turned away.

Service discipline:

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


'Ihe number of parallel servers in the system.


'Ihe number of servers in series within a given channel.

Queuing systems are characterized by the arrival distribution of the calling population, the service distribution, the number of services, the number of phases, the service discipline, and system capacity.

of the process. Table 5.7 describes the basic components of a queuing model. The arrival distribution or calling population defines the first characteristic. The second is the probability distribution of service provided to the calling population. The third is the system capacity. Some systems cannot accommodate all arrivals in a specific period. An example is a retail store with limited parking or the drive-through window at a bank that only allows a specific number of automobiles to wait in line. The fourth system characteristic describes the service discipline. Some process systems allow first come, first served prioritization (i.e., first in, first out, or FIFIO), whereas others use a different service discipline. The number of channels in the system refers to the total available parallel servers in the model. An example would be a bank that has several associates who provide service to customers waiting in line. The number of phases refers to the number of subsequent operations past the first sever in a channel.

A concise notation was developed to make their descriptions easy to understand and interpret because there are different types of queuing models and each is specified by the characteristics of the system being


Modified Kendall’s Notation (a/b/c): (d/e/f)



Arrival and service distributions:

M = exponentially distributed Ek = Erlang type - к distributed D = deterministic or constant G = any other distribution

Service discipline:

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

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, and e = the system’s capacity; and a final addition by H.A. Taha: f = size of the calling population (i.e., infinite or finite).

modeled. This notation was developed by Kendall and is shown in Table 5.8 along with modifications. Table 5.8 lists common probability distributions used to describe the arrival and service patterns that occur within a system. Table 5.9 summarizes our queuing discussion and lists the important queuing model characteristics. 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 collect data for the arrival and service rates


Queuing Model Characteristics

Calling Population

Service Discipline

1. Infinite distribution

1. Deterministic (constant) service

2. Finite distribution

2. Distributed pattern of service

3. Deterministic arrivals

3. Service rules (FIFO, LIFO, SIRO, POS, GSD)

4. Distributed arrivals

4. Single phase

5. Balking or reneging allowed?

5. Multiple phases

6. Single channel

7. Multiple channels

FIFO = first in, first out; LIFO = last in, first out; SIRO = service in random order; POS = prioritization of service; GSD = general service discipline.


Common queuing models.

and the other system characteristics to build the model and understand the expected process performance. The characteristics of these models are shown in Table 5.10 using Kendall notation. This notation succinctly represents the key characteristics of each model based on the underlying process. There are other types of queuing models that can be matched your process. A literature search is the best way to find the model that matches your team’s requirements.

The (M/Ml) model can analyze processes characterized by Poisson arrival and exponential service distributions, a single channel with a first- come-first serve (FIFO) service discipline. An example would be waiting in line at a ticket office where there is one server. The (M/M/k) model is used to analyze processes with several parallel servers. Examples would be waiting lines in supermarkets and banks when there are several associates processing transactions. The (M/G/k) model, if modified with a capacity constraint, is used to analyze processes with a finite number of arrivals allowed into the system. An example would be a website designed to handle a limited number of incoming transactions. The fourth model (M/M/l) is applied to situations where the calling population that requires

TABLE 5.10

Characteristics of the Four Models

Model Type











Single channel

Poisson arrivals, infinite calling population

Exponential service distribution





Poisson arrivals, infinite calling population

Exponential service distribution





Poisson arrivals, infinite calling population (capacity constrained)

General service distribution



Single channel

Poisson arrivals, finite calling population

Exponential service distribution


FIFO = first in, first out.

service is small (finite). An example would be a repair shop servicing a limited number of on-site machines.

An example of the (M/M/l) queuing model is shown in Table 5.11. Its formulas are easy to calculate manually, although other models may have more complicated calculations. In this example, customers arrive at an average rate of 20 per hour. The arrival rate fluctuates with customers arriving faster or slower than 20 per hour. The service rate is 25 customers on average per hour. On average there is capacity to provide service. A requirement of queuing models is that the average service rate must exceed the average arrival rate. The variation of arrival rates will require some customers to wait for service sometimes; alternatively, if arrival rates are low, then servers will sometimes be idle.

To build a model, arrival and service rates are estimated empirically with check sheets, automatically by software specialized applications that measure transactions and transformations with business rules between systems or using historical records. Integrating data into a queuing model helps show where capacity should be added and how to set the system’s

TABLE 5.11

Example of a Simple Queuing Model (M/M/l)



1. Arrival rate into system (X)

X = 20 per hour

2. Average units serviced (g)

g = 25 per hour

3. System utilization factor (A./pi) note: X/g < 1

X/[l = 0.80 = 80%

4. Average number of units in system (L)

L = Lq + (A/g) = 4.0 units

5. Average number of units in queue (Lq)

Lq = A2/[g(g—A)] = 3.2 units

6. Average time a unit spends in system (W)

W = Wq +(1/g) = 0.20 hours

7. Average time a unit waits in queue (Wq)

Wq = Lq/ X = 0.16 hours

8. Probability of no units in system (P0)

P0= 1—(A/g) = 0.20 = 20%

9. Probability arriving unit waits for service (Pw)

pw = X/[i = 0.80 = 80%

10. Probability of n units in the system (P„)

P„ =(A/g)n

In this example, customers arrive at a repair shop at an average rate of X = 20 per hour; the average service rate is 25 customers per hour. Assume a Poisson arrival distribution, an exponential service distribution, a single-channel/FIFO service discipline, no maximum on the number in the system, and an infinite calling population. FIFO = first in, first out.

rules to minimize customer waiting time and system cost while achieving service levels. Queuing models can be combined with marketing research to build competitive service levels and waiting times by segment to ensure high customer satisfaction and high operational efficiency.

Table 5.12 shows another advantage of queuing analysis. The units of measure are in hours. As an example, for the last analysis, 49 customers arrive each hour and the system can service 50 customers. The average

TABLE 5.12

Queuing Analysis Study of Capacity Utilization




















Average Time in System









































As the systems capacity utilization approaches 100%, waiting time significantly increases. Assumption include a single-channel queue, a Poisson arrival rate, and an exponential service rate.

waiting time is 0.277 hours or 16.62 minutes. As the utilization of a system increases, the average waiting time for service rapidly increases. In a service pool like a call center, capacity can easily be increased to match demand. This may not be possible if servers are few and highly skilled (i.e., cannot be replaced easily or others quickly trained). But there are operational strategies that efficiently add capacity with high utilization. In manufacturing, one strategy is to use low-cost machines in parallel to each other, with some idle during periods of low demand. These idle machines are activated to meet demand rather than fully utilized regardless of demand. The old paradigm was to use expensive machines and keep them running at all times to attain high utilization or production efficiency. This built excess inventory. An analogous situation would be using low-cost workers to complete smaller and standardized work rather than more complicated work. Another example would be a call center that uses technology to transfer incoming customer calls that exceed local capacity to other call centers that have excess capacity at the time (i.e., level-loading demand across a global system).

Another type of analytical algorithm useful for designing and optimizing a process is linear programming (LP) and its various models. Table 5.13 lists several linear programming models, but there are many others. LP models minimize or maximize an objective function. Maximization or minimization (i.e., optimization) is constrained relative to resource scarcity, minimum service levels, and many other factors. Optimization objectives include maximizing profits, minimizing cost, maximizing service levels, or maximizing production throughput. Table 5.14 shows the basic components of an LP model to build a supply

TABLE 5.13

Linear Programming Applications


1. Maximizing service productivity.

2. Minimizing network routing.

3. Optimizing process control.

4. Minimizing inventory investment.

5. Optimizing allocation of investment.

6. Optimizing product mix profitability.

7. Minimizing scheduling cost.

8. Minimizing transportation costs.

9. Minimizing cost of materials mixtures.

TABLE 5.14

Linear Programming Characteristics

What is Linear Programming?

1. An LP algorithm attempts to find a minimization maximization or solution when decisions are made with constrained resources as well as other system constraints. As an example, supply chain optimization problems require matching demand and supply when supply is limited, and demand must be satisfied. An LP problem is comprised of four major components:

1. Decision Variables within analyst’s control...When and how- much to order ...When to manufacture...When and how much of the product to ship.

2. Constraints placed on the levels or amounts of decision variables which can be usedin the final solution...Examples are: Capacity toproduceraw materials or components... Production can only run for a specified... number of hours... A worker can only work so much overtime ...A customer’s capacity to handle and process receipts.

3. Problem objective relative to minimization or maximization. Examples include maximizing profits, minimizing cost, maximizing service levels andmaximizing production throughput.

4. Mathematical relationships between the decision variables, constraints, and problem objectives.

When do we have a solution to a linear program?

1. Feasible Solution - Satisfies all the constraints of the problem or objective function.

2. Optimum Solution - The best feasible solution, relative to die decision variables and their levels, that achieves the objective of the optimization problem. Although there maybe many feasible solutions, there is usually only one optimum.

chain model. It includes decision variables that can be varied for optimization, including when and how much to order, manufacture, or ship through logistical systems. Constraints are limitations placed on the decision variables such as available capacity, manufacturing scheduling sequence, materials or components, minimum sales level, and others. An LP model is used to evaluate relationships between decision variables and their constraints relative to the model’s objective. An optimum solution will be the best feasible solution relative to the levels of the decision variables that achieve the objective of the optimization problem, which is either minimization or maximization without violating the constraints. Although there may be several feasible solutions, there is usually only one optimum solution.

Figure 5.14 shows how an LP model is mathematically constructed. The objective function represents the goal of the optimization. Each decision variable Xt is weighted by an objective coefficient C,. The optimization


Basic linear planning model formulation.

of the objective function is constrained by the minimum or maximum amount of resources that can be used in the final solution. Each decision variable in a constraint is weighted by a coefficient showing its relative contribution to optimization. The right-hand side (RHS) of each constraint can be of three types: less than or equal to, equal to, and greater than or equal to. Less than or equal to implies the resources when used in combination cannot exceed a maximum. These are usually applied to limited material and labor resources. Equal to imply the combination of decision variables must exactly equal a number. In greater than or equal to equations, the combination of decision variables cannot fall below minimum value. These are usually associated with minimum demand that must be satisfied. In standard LP models, all decision variables are constrained to be positive (i.e., X; > 0).

The example shown in Figure 5.15 involves a simple transportation network consisting of three manufacturing facilities and four distribution centers. This is special type of LP algorithm called the transportation model. The most common objective of a transportation model is to minimize transportation costs between manufacturing facilities and distribution centers. This example shows two constraints: the maximum available material that can be shipped from a given manufacturing facility, and the demand of each distribution center for the material. If the material supply and required demand do not balance, then “dummy” manufacturing facilities or distribution centers are incorporated into the model to balance supply and demand constraints. The problem shown in Figure 5.15 is analyzed using Excel’s “Solver” algorithm, but other software can be used for analysis. The optimum solution is also shown in the “From/To” matrix with the shipment costs of the facility and distribution center (DC) combinations. As an example, the per-unit cost from Facility 1 to DC 1 is $10, and the maximum available supply from Facility 1 is 100 units. The demand of each distribution center is shown as DC 1 = 150 units, DC 2 = 200 units, DC 3 = 50 units, and DC 4 = 200 units. The optimum solution is shown in the Candidate Solution matrix. In this solution, Facility 1 ships 50 units to DC 3 and 50 Units to DC 5. The total cost of the optimum solution is shown in the Cost matrix as $3,750. A general form of the model is shown at the bottom of Figure 5.15. LP models have proven useful in process workflow modeling and analysis in many diverse fields of business and science.


Transportation system example. DC = distribution center.

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