Ajay Shenoy
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Session-06

The Memoryless Property: Why Past Debugging Time Tells You Nothing

You have seen Poisson count events over an interval. Now consider the gap between two consecutive events — how long do you wait? A distributed key-value store serves reads from two independent replicas; the response time of each replica is a continuous random variable called Exponential. What is the difference between a discrete count and a continuous waiting time, and why does the same parameter $\lambda$ appear in both?

graph LR TIME["Time axis"] --> E1["Event 1"] E1 -->|"T1~Exp(λ)"| E2["Event 2"] E2 -->|"T2~Exp(λ)"| E3["Event 3"] E3 -->|"T3~Exp(λ)"| E4["..."]

The inter-arrival time between Poisson events follows an Exponential distribution — the two are two sides of the same coin. Poisson counts events in a fixed window; Exponential measures the gap between them. Both share parameter $\lambda$. The next segment derives the Exponential CDF $F(t) = 1 - e^{-\lambda t}$ directly from the Poisson PMF.

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P99 Latency is 4× the Mean — The Stat That Should Scare Every Engineer

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