# Time-Varying M/M/1 Queue: Maximum Queue and Clearance Time ## Problem Recap **Service rate:** \(\mu 5/\text{min}\) (constant) - **Arrival rate (\(\lambda(t)\)):** - \( \leq t < 30:\)  \(\lambda = 2/\text{min}\) - \(30 \leq t < 90:\) \(\lambda = 6/\text{min}\) - \(t \geq 90:\)    \(\lambda = 2/\text{min}\) - **Initial condition:** \(Q() = \) (empty queue) - **Fluid approximation:** - \(\frac{dQ}{dt} = \lambda(t) - \mu\) when \(Q > \) - If \(Q(t)\) would go negative, set \(Q(t) = \) ("reflect at zero") - **Goal:** 1. Find maximum expected queue length (\(Q_{\text{max}}\)) 2. Find clearance time (\(t_{\text{clear}}\)), when queue returns to zero after overload --- ## 1. Fluid Model Solution ### Interval 1: \( \leq t < 30\) \[ \lambda = 2 < \mu = 5 \implies \frac{dQ}{dt} = -3 \] - Since the queue is empty and arrivals are less than service, \(Q(t) = \) (cannot go negative). --- ### Interval 2: \(30 \leq t < 90\) \[ \lambda = 6 > \mu = 5 \implies \frac{dQ}{dt} = 1 \] - At \(t = 30\), \(Q(30) = \) - For \(30 \leq t < 90:\) \[ Q(t) = Q(30) + (t - 30)(\lambda - \mu) = (t - 30) \times 1 \] - At \(t = 90:\) \[ Q(90) = (90 - 30) \times 1 = 60 \] --- ### Interval 3: \(t \geq 90\) \[ \lambda = 2 < \mu = 5 \implies \frac{dQ}{dt} = -3 \] - \(Q(90) = 60\) - For \(t \geq 90:\) \[ Q(t) = Q(90) - 3(t - 90) \] - Queue clears when \(Q(t_{\text{clear}}) = \): \[ = 60 - 3(t_{\text{clear}} - 90) \] \[ 3(t_{\text{clear}} - 90) = 60 \implies t_{\text{clear}} - 90 = 20 \implies t_{\text{clear}} = 110 \] --- ## 2. Results Summary ### 2.1 Maximum Expected Queue - **Occurs at end of overload (\(t = 90\) min):** \[ Q_{\text{max}} = Q(90) = 60 \] ### 2.2 Clearance Time - **Queue empties at:** \[ t_{\text{clear}} = 110 \text{ minutes} \] --- ## 3. Queue Trajectory (Piecewise Linear) | Interval | \(\lambda\) | \(\frac{dQ}{dt}\) | \(Q(t)\) formula | |------------------|--------------|-------------------|----------------------------------| | \( \leq t < 30\) | 2/min | | \(Q(t) = \) | | \(30 \leq t < 90\) | 6/min | 1/min | \(Q(t) = t - 30\) | | \(90 \leq t < 110\)| 2/min | -3/min | \(Q(t) = 60 - 3(t - 90)\) | | \(t \geq 110\) | 2/min | | \(Q(t) = \) | --- ## 4. Visualization ``` t (min): 30 90 110 Q(t): ───── /───── 60 \───── (builds) (drains) ``` - [Graph illustration: Queue remains at until \(t=30\), then rises linearly to 60 at \(t=90\), then falls linearly to at \(t=110\).] --- ## 5. Key Takeaways - **Maximum queue:** \(Q_{\text{max}} = 60\) at \(t = 90\) min. - **Queue clears:** \(t_{\text{clear}} = 110\) min. - **Piecewise linear:** Fluid model is simple to compute for stepwise \(\lambda(t)\). - **Reflecting at zero:** No negative queue due to underloaded intervals. --- **Note:** This is the *fluid* (deterministic) expected queue; real queues will fluctuate around this due to stochastic arrivals.