Chapter 8: Problem 7

\text { What happens to CT, WIP, } C T_{q} \text { and } \mathrm{WIP}_{q} as the arrival rate \(r_{a}\) approaches the process rate \(r_{e} ?\)

Short Answer

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Question: Describe the behavior of Cycle Time (CT), Work in Progress (WIP), Queue Cycle Time (CT_q), and Queue Work in Progress (WIP_q) as the arrival rate (r_a) approaches the process rate (r_e). Answer: As the arrival rate (r_a) approaches the process rate (r_e), Work in Progress (WIP) remains stable, Cycle Time (CT) remains consistent, Queue Cycle Time (CT_q) decreases, and Queue Work in Progress (WIP_q) also decreases.

Step by step solution

Understand Little's Law

Little's Law states that WIP = CT × r_a. It describes the relationship between the number of items in a system (WIP) and the time it takes to complete an item (CT) based on the arrival rate (r_a) of those items. Similarly, the law can be applied to queues as WIP_q = CT_q × r_a.

Work In Progress (WIP) as r_a approaches r_e

As r_a gets closer to r_e, it means that items are entering the system at a rate that is almost equal to the rate at which they're being processed. As a result, the WIP (number of items in the system) will become more stable and consistent since items will not pile up or significantly decrease.

Cycle Time (CT) as r_a approaches r_e

Since WIP is stable and consistent when r_a approaches r_e, the CT (time taken to process an item) will also remain consistent, as WIP = CT × r_a according to Little's Law. This indicates that the overall time taken to complete an item will not change drastically as the arrival rate gets closer to the processing rate.

Queue Cycle Time (CT_q) as r_a approaches r_e

In the case of queues, as r_a approaches r_e, it means that the items are entering the queue almost at the same rate as they are being processed, which results in a smaller overall queue. Therefore, the CT_q (time items spend in the queue) will decrease since the items will not have to wait as long in the queue before being processed.

Queue Work In Progress (WIP_q) as r_a approaches r_e

When the arrival rate is close to the processing rate, it results in a smaller queue as mentioned earlier. As a consequence, the WIP_q (number of items in the queue) will also decrease, since fewer items will be waiting to be processed at any given time. This can be understood through the relation: WIP_q = CT_q × r_a, which states that as CT_q decreases, WIP_q will also decrease. In summary, when the arrival rate (r_a) approaches the process rate (r_e), WIP remains stable, CT remains consistent, CT_q decreases and WIP_q also decreases.

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\text { What happens to CT, WIP, } C T_{q} \text { and } \mathrm{WIP}_{q} as the arrival rate \(r_{a}\) approaches the process rate \(r_{e} ?\)

Short Answer

Step by step solution

Understand Little's Law

Work In Progress (WIP) as r_a approaches r_e

Cycle Time (CT) as r_a approaches r_e

Queue Cycle Time (CT_q) as r_a approaches r_e

Queue Work In Progress (WIP_q) as r_a approaches r_e

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