# Why should you calculate the waiting time?

Most people don’t like to wait, be it in a doctor’s waiting room, a shop, or an airport. So, understanding the science of waiting times and finding methods to tackle the waiting time problem is of interest for many.

## Terminology

In order to communicate clearly about the science of waiting times, we have to introduce some terms:

• SERVER: a server refers to a unit (either a human being, a machine, or a combination of both) serving a customer. A server can be a doctor seeing a patient, a phone operator, the combination of a cashier at a cash register in a supermarket, etc. In the example, there are 5 persons waiting, while 3 servers are busy.
• SERVICE TIME: the time it takes to serve a customer. In real life situations, service times usually vary over time. For instance, in a supermarket, the service time depends on the number of items the customer bought. In order to describe service times measured in a certain time period, we need to compute two parameters:
• Average service time
• Standard deviation (of service times)
• ARRIVAL TIME: the arrival time refers to the moment when customers are arriving in the waiting area. So it is a point in time, not a duration. For practical purposes, we need to calculate the inter-arrival time, i.e. the time interval between arrival of two customers. Again, we need to compute 2 parameters:
• Average inter-arrival time
• Standard deviation (of inter-arrival time)

## Fundamental dilemma

There is a fundamental contradiction between optimizing the fraction of time that the servers are busy and optimizing waiting times.

Moreover, waiting time are increasing more rapidly if there is more VARIABILITY in the system. Variability can be either in the service time or in the arrival time, or in both.

Of course, this contradiction is rather unfortunate in many circumstances. Usually, organizations want their servers to be busy (for economical reasons), while they don’t want their customers to wait too long (customer satisfaction).

Understanding the science behind this fundamental dilemma helps organizations to think about possible solutions and their consequences.

## Formula

Waiting time formula for multiple (m) servers:
$$T_q = \left({p \over m}\right) \times \left({utilization^{\sqrt{2(m+1)}-1} \over 1-utilization}\right) \times \left({CV_a^2 + CV_p^2 \over 2}\right)$$
With
$$Utilization \, = \frac{p}{m * a}$$
and
$$Coëfficient\,of\,variation \, (CV) \, = \frac{Standard \, Deviation \,(\sigma)}{Average \, \overline x}$$
$$Average \, \overline x = \frac{1}{n} \sum_{i=1}^n x_i$$
$$Standard \, Deviation \,(\sigma) = \sqrt{ \frac{1}{N} \sum_{i=1}^N (x_i -\overline x)^2}$$

#### Abbreviations & terminology

• p: process time, activity time or service time
• a: inter-arrival time
• m: number of servers
• Utilization: fraction of time that the server(s) is (are) busy
• CVa: coefficient of variation of inter-arrival times (standard deviation / mean)
• CVp: coefficient of variation of service times (standard deviation / mean)

#### Calculate the average waiting time

= ...

For more details, see “Matching Supply with Demand: An Introduction to Operations Management” - 4th Edition Gerard Cachon & Christian Terwiesch

## Categories of solutions

When waiting times are too long, there are several theoretical ways to overcome the issue:
1. Increasing the number of servers. For instance, supermarkets are typically monitoring waiting times (or the number of customers in the waiting area) and temporarily increasing the number of cashiers when needed.
2. Segmenting the customer base. By separating specific groups of customers, waiting time can be reduced for at least some of them. Examples are a quick checkout cash register in a supermarket (for people with a low number of items), “business class” in airports, etc.
3. Inflow restriction or regulation. In order to reduce variability of inflow, customers can be allowed into the system at specified time points only. For instance, in hospitals, patients can be given appointments instead of free access.
4. Standardization and grouping of procedures. The more service procedures can be standardized, the lower the variability in service times, and the lower the effect on waiting times. By grouping certain categories of procedures, and combining this with inflow regulation, waiting times can be optimized. In the extreme case, it may be a strategy to deliver only services that can be fully standardized. As a theoretical example, a hospital could focus on routine surgical procedures and refuse complicated cases (“focused factories”)

## Practical approach

A practical approach to tackle waiting time problems is presented below. It consists of a few simple steps.

#### Step 1. Do some simple measurements to asses the current situation

1. You have to measure 2 things only: service times(1) and arrival times(2)
2. Example. Imagine that you run a radiology department in a hospital and you examine 100 patients per day in a busy ultrasound department, using 3 “servers” (ultrasound machine and radiologist). Patients come without appointment. What you have to measure is:
• (1) the exact duration each exam (time between entering and leaving the exam room, thus medical + non-medical time)
• (2) the exact point in time when each patient arrives
3. Ideally, in order to have a good sample, you do this during one week. In order to obtain all measurements needed, you need at least one FTE to coordinate this task.

#### STEP 2. Calculate the number that can be used for data input

1. From the measurements obtained in step 1, you need to calculate the following:
• Service times: average
• Service times: standard deviation
• Inter-arrival times: average
• Inter-arrival times: standard deviation

#### STEP 3. Input your numbers into the input fields

1. Besides the info on service times and inter-arrival times, you also have to enter the number of servers in the input fields.
3. The output field gives you the calculated average current waiting time based on the Cachon & Terwiesch formula.

#### STEP 4. Design a strategy for improvement and asses its effect on waiting times

1. The exercise starts with a management decision: what do we want? What is acceptable in terms of average waiting times?
2. The number of possible strategies to achieve a specified goal is large, and multiple combinations exist, depending on the context. Just an example here to illustrate the approach.
3. Sample strategy: a very simple and straightforward strategy could be to regulate inflow by accepting patients at specific time slots only. By doing this, the standard deviation of inter-arrival times (theoretically) becomes 0, and waiting times will become lower, even when examination times and average inter-arrival times stay unchanged. As a next step, the length of the time slots (for patient arrival) can be varied and the effect on waiting times can be assessed (by using the model). In this way, optimal time slot durations can easily be designed for different types of exams.

#### STEP 5. Implement the chosen strategy and repeat the process if needed

1. The model gives you a scientific base to propose and implement operational changes (and to convince stakeholders, if needed).
2. Once the changes have been implemented, the predicted effect on waiting times may optionally be confirmed by repeating the measurements described in step 1. This may also provide a basis for further refinements if needed.

## Authors

Any questions? Contact me

#### Willem Van Hoe

Master in Medicine student at KU Leuven, Belgium

Master in Management student at IE Business School, Spain

Any questions? Contact me