A system is monitoring the average response time of a web server. You are given the following information. Your task is to calculate the updated cumulative average response time after the newest request is processed. Provide only the final numerical value in milliseconds.

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A cumulative average can be calculated using a recursive formula. This means the average at the current step can be computed based on the value of the average from the preceding step and the current data point, which is more efficient than recalculating the sum over the entire history at each step.

Recursive Formula for Cumulative Average

The cumulative average for the memory component can be calculated efficiently using a recursive formula. The memory state at the current step, $$\mathrm{Mem}_i$$, is computed based on the current key-value pair $$(\mathbf{k}_i, \mathbf{v}_i)$$ and the memory state from the previous step, $$\mathrm{Mem}_{i-1}$$. The recursive formula is expressed as: $$\mathrm{Mem}_i = \frac{(\mathbf{k}_i, \mathbf{v}_i) + i \cdot \mathrm{Mem}_{i-1}}{i + 1}$$ This approach prevents the need to recalculate the sum of all historical key-value pairs at each step, significantly increasing computational efficiency.

Recursive Formula for Memory as a Cumulative Average

A real-time analytics system tracks the average score for a popular online game. After 1,000,000 games have been played, the system has calculated a cumulative average score. When the 1,000,001st game score is recorded, which of the following methods for updating the average is the most computationally efficient and mathematically correct?

Real-time Performance Monitoring

A system for monitoring network latency has processed 99 data packets and calculated a cumulative average latency of 250 milliseconds (ms). The 100th data packet arrives and has a latency of 350 ms. Using an efficient recursive method that does not require summing all 100 values, calculate the new cumulative average latency. Show the primary calculation step.

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