# Example if one customer can be served in 10 minutes

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Example: if one customer can be served in 10 minutes on average, then 0 min/customer , customers/min customers/hour – we can serve 6 customers per hour Watch out: in some situations “service rate ()” is given; in others it is “average service time ()” that is given.
Capacity, Demand, and Flow Rate If we have identical servers, each with service rate , then the capacity of our system (total service rate) equals . Capacity = Arrival rate of customers, , is the demand for service. Arrival rate = Demand = Flow Rate is the actual number of customers served per unit of time. Flow Rate = min(Demand, Capacity) = min(, ) If Demand Capacity (), capacity is sufficient to satisfy demand. In this case, Flow rate = Demand = . All customers can be served. If Demand Capacity (), capacity is insufficient for the demand. In this case, Flow rate = Capacity = . Some customers will not be served. All queuing models we will deal with assume: (i.e., sufficient capacity).
Steady State Vs Transient A system is in steady state (i.e., not in a transient state) & steady-state analysis is valid, if: System is stable (arrival rate < service rate, i.e., sufficient system capacity). Parameters are constant for period under analysis. System has been running long-enough for effects of initial conditions to have disappeared. Most queuing results pertain to steady state; else resort to simulation.
Graph of number of customers in system over time No. of customers 2 1 1 2 3 4 5 6 7 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 Time Find: Avg. Flow Rate; Inter-arrival Time = Avg. Service Rate; Average Service Time = W = Avg. Time in System; = Avg. Time Waiting in Queue L = Avg. # of Units in System; = Avg. # of Units Waiting in Queue = Utilisation Capacity = A Deterministic Queue
Graph of number of customers in system over time No. of customers 2 1 1 2 3 4 5 6 7 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 Time Find: Avg. Flow Rate = 1/3; Inter-arrival Time = 3; Avg. Service Rate = 1/2; Average Service Time = 1/=2; W = Avg. Time in System = 2; = Avg. Time Waiting in Queue = 0; L = Avg. # of Units in System = 2/3 ; = Avg. # of Units Waiting in Queue = 0; = Utilisation = 2/3; Capacity = 1/2; A Deterministic Queue
Little’s Law: For any system/process operating in steady state Examples On average, customers visit the bank every hour and spend 12 min 0.2 hours, on average, in the bank. How many customers are there in the bank, on average? Avg. # of customers in a bank On average 6,500 undergraduate students study at AUB. Average duration of studying is 3.5 years. How many students join AUB per year, on average? Avg. Flow Rate . System/process Arrival Departure Average Number of Units in the System Average Flow Rate Average Time in System ? = ? × ? [units of flow] [units of flow / units of time] [units of time]
Little’s Law A queuing system consists of a queue and a service.

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• Fall '19
• Normal Distribution, Probability theory, Exponential distribution, Poisson process, Queueing theory

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