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# Tag Archives: Inter-Arrival Time

## Reducing waiting time by pooling demand

By **pooling demand**, the inter-arrival times are shortened and thus the specific demand goes up (which is intuitive, since pooling demand basically means combining different demand streams). While the utilization rate is not effected by demand pooling, the waiting time is shortened because some inefficiencies (idle time at station A while station B is overwhelmed) are eradicated. However, pooling more and more resources together also decreases the overall efficiency once the demand is met. Therefore, **companies need to find a viable balance between efficiency and responsiveness**.

What main benefits and costs are connected with pooling in the context of waiting time?

- Pooling assumes total flexibility (Spanish call center agents will not be able to answer to German customers, even if the call center company decided to pool all calls together).
- Pooling increases the complexity of the workflow, since demands needs to be shifted between resources who might be locally apart (e.g. two hospitals or two plants).
- Pooling interrupts the continuity of interaction between the flow unit (customer) and the resource (worker) and can thus hurt the customer experience because customers will not want to see a different physician or a different financial consultant on every separate visit.

These lecture notes were taken during 2013 installment of the MOOC “An Introduction to Operations Management” taught by Prof. Dr. Christian Terwiesch of the Wharton Business School of the University of Pennsylvania at Coursera.org. |

## Variability in demand and processing

**Variability in demand**

If there is more demand than capacity, the implied utilization rate rises above 100%, which makes waiting time unavoidable. The more interesting cases are those, in which there is waiting time even though the implied utilization rate is below 100%. Such waiting time stems from demand variability generated by the somewhat **random nature of most demand processes**, e.g. many customers showing up at once at some point in time and no customers showing up at all at some other point in time.

In order to calculate with **demand variability**, we need to define **arrival time**, **inter-arrival time** and **average inter-arrival time**. The self-explanatory arrival time is defined as the time, when customers arrive at a business. The inter-arrival time is thus defined as the time between subsequent customer arrivals. If demand is random, both the arrival times and the inter-arrival times will be drawn from an underlying statistical distribution (often a Poisson distribution). The average inter-arrival time is usually denoted with a.

Another important parameter is the **coefficient of variation** of the arrival time, which is calculated as the **standard deviation over the mean** and is denoted as Cv_a. The coefficient of variation is a way to measure the standard deviation against the mean of the distribution. This is useful, because the standard deviation itself is not really a good measure for variability, since it does not express whether a 10 minute deviation is a lot or not.

If the inter-arrival times are drawn from an Poisson distribution, the coefficient of variation is always 1. This knowledge can be used to calculate other parameters considering this formula:

*Cv_a = standard deviation (of inter-arrival times) / mean (of inter-arrival times) = 1*

**Variability in processing**

Variability is not limited to the demand process, but also occurs in processing. The calculations are basically the same, with the average processing time being denoted with p and the coefficient of variation being denoted with Cv_p. The coefficient of variation can be seen as a measure of the degree to which a work process has been standardized.

These lecture notes were taken during 2013 installment of the MOOC “An Introduction to Operations Management” taught by Prof. Dr. Christian Terwiesch of the Wharton Business School of the University of Pennsylvania at Coursera.org. |