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I have a real-life problem from work that we haven't been able to figure out. None of us have advanced CS background.

Rate R of message arrival to the system from a client is 7/10s.

25 workers do little processing, then send the messages to Service.

The time T is 3s to: send message to Service, Service to process, and to receive reply from Service. Let S = 1/T

When R > S, the system queues messages to the 25 workers. We don't know the algorithm but can assume there is a queue for each worker.

  1. How can we find the length of time Q a message waits in a queue, over all workers on average, after M messages?

  2. Does the "end to end" time for a message to arrive to system from client, process the message (including Service time), and return to client increase without bound, or does it reach a steady state?

From the answer given on Little's Law in Queueing Theory: How to estimate steady-state queue length for single queue, N servers?

Then Little's Law to answer (1), we have $L = {\lambda }W$.

In this case,

$Q = R \cdot T = \frac{7}{10} \cdot 3 = \frac{3}{10}$

But I'm not very confident in this answer and looking for help,

Thanks!

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