The average total time in the system (including time to be served) is calculated using which of the following equations for an infinite queue model?单项选择题
A
Wq= Lq λ Lq λ
B
Lq= Ls λ Ls λ
C
Ws= Ls λ Ls λ
D
Ls= Ws λ Ws λ

Question

The average total time in the system (including time to be served) is calculated using which of the following equations for an infinite queue model?

BlackTom AI · Accepted Answer

Little's Law states L = λW, so the average number in a system equals arrival rate times average time spent; applying this to the entire system gives Ws = Ls / λ.

1) Relevant knowledge points and concepts
- Little's Law: For a stable system, average number in system L = λ × average time in system W. It holds under mild conditions (work-conserving, stationary) and applies to infinite-capacity queues (M/M/1, M/G/1, etc.).  
- Distinction between system and queue: Ls and Ws include customers in service; Lq and Wq refer only to the waiting line.

2) Complete reasoning process (step by step)
- Let T be a long time interval and N be the number of arrivals during T. The arrival rate λ ≈ N/T.  
- Let Si be the time customer i spends in the system (waiting + service). The total customer-time spent in system over T is Σ Si.  
- Average number in system Ls = (total customer-time in system) / T = (Σ Si) / T.  
- Average time in system Ws = (total customer-time in system) / N = (Σ Si) / N.

3) Intermediate algebraic steps
- From definitions: Ls = (Σ Si)/T and Ws = (Σ Si)/N.  
- Divide these: Ls / Ws = (Σ Si / T) / (Σ Si / N) = N / T.  
- But N / T = λ, therefore Ls = λ Ws.  
- Rearranging gives Ws = Ls / λ.

4) Why the correct answer is correct
- The formula Ws = Ls / λ directly follows from Little's Law applied to the entire system (including service). It correctly relates average system population to average time per customer and arrival rate.

5) Why other options are incorrect
- Option 1 (Wq = Lq / λ) is a true relation but refers only to waiting time in queue, not total time including service, so it does not answer this question.  
- Option 2 (Lq = Ls / λ) mixes queue and system quantities and is dimensionally incorrect.  
- Option 4 (Ls = Ws / λ) inverts the correct relation; the correct form is Ls = λ Ws, not Ls = Ws / λ.

6) Conclusion
- Therefore the correct equation for average total time in system is Ws = Ls / λ, which matches the provided answer.

The average total time in the system (including time to be served) is calculated using which of the following equations for an infinite queue model?单项选择题

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