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# 1 square root law in mm queue the number of servers is

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Unformatted text preview: needed in month t • zt # to hire as trainees in month t Long Term Planning 0 zt = max @0, ⌘t+⌧ t+⌧ 1 X yj (1 r) j =t ⌧ (j t) A at time t, plan for time t+τ sometimes, what you have is more than what you need time length from j to t+τ-1 applicable in a wider range! 1 Square Root Law • In M/M/∞ queue, the number of servers is Poisson with parameter R=λ/µ. • Poisson(R) can be approximated by Normal(R,R) ⇡1 ✓ NR p R ◆ Standard normal(0,1) distribution function Square Root Law • Now if delays are not prevalent, the number of busy servers in a ﬁnite server queue can be approximated by the number of inﬁnite server queue. • So P(wait) = P(# busy serves >N ) ﬁnite inﬁnite ⇡ P(# busy serves >N ) ✓ ◆ NR computable p ⇡1 R Square Root Law • To decide the number of servers N ✓ NR p R ◆ =1 NR p = R P(wait) N =R+ = P(wait) P(wait) 1 (1 P(wait) P(wait)) p R Part 8: Process Flow Analysis Process ﬂow diagram I Processing times Capacity Capacity per hour III 37 Resource II 46 37 I II III 37 46 37 [sec/cust] 0.02703 0.02174 0.02703 [cust/sec] 97.29730 78.26087 97.29730 [cust/hour] Process capacity Demand 78.26087 [cust/hour] 60 60 60 60 [cust/hour] Utilization 0.617 0.767 0.617 Cycle time 60 60 60 [sec] Idle time 23 14 23 [sec] Total idle time 60 [sec] Labor content 120 [sec] Labor utilization 0.667 Basic deﬁnitions • Processing times: how long does the worker spend on the task? • Capacity = 1/processing time: • how many units can the worker make per unit of time • If there are m workers at the activity: Capacity = m/activity time • Bottleneck: process step with the lowest capacity • Process capacity: capacity of the bottleneck • Flow rate = Minimum{Demand rate, Process Capacity) • Utilization = Flow Rate / Capacity • Flow Time: The amount of...
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