End-to-end packet delay is an important metric for performance and Service Level Agreements (SLAs).Queuing Networks l2 l1 0.3 0.7 l6 l3 l4 0.5 l5 0.5 l7īridging Router Performance and Queuing TheorySigmetrics 2004 Slides by N. If any given item has a probability P1 of “leaving” the stream with rate l1:.two poisson streams with arrival rates l1 and l2:.Past state does not help predict next arrival.Poisson process = exponential distribution between arrivals/departures/service.The number of buffers so P(overflow) is mean number of packets in the gateway?.What is the probability of n packets in the gateway?.
Assuming exponential arrivals & departures:.Solving for P0 and Pn 1:, ,, 2:, , (geometric series) 3: 5: 4: In general:Įquilibrium conditions l l l l n-1 n n+1 m m m m 1 2 inflow = outflow 1: 2: stability: 3: Finding L is hard or easy depending on the type of system.W: average waiting time in whole system.m: Service rate of the server (output link).l: Arrival rate of jobs (packets on input link).Goal: A closed form expression of the probability of the number of jobs in the queue (Pi) given only l and m.General arrival and service distributions, 3 servers, 17 queue slots (20-3), 1500 total jobs, Shortest Packet First.Exponential arrivals and service, 1 server, infinite capacity and population, FCFS (FIFO).M/M/1 is the simplest ‘realistic’ queue.Total Capacity (infinite if not specified).Probability distribution packet is serviced in time t.Probability of a new packet arrives in time t.First three typically used, unless specified.
Queuing System l l m Server Queue Model Queuing System Strategy: Use Little’s law on both the complete system and its parts to reason about average time in the queue Queuing System Server System What is average waiting time before and in the tunnel?.Observe 32 cars/minute depart over a period where no cars in the tunnel at the start or end (e.g.Observe 120 cars in front of the Lincoln Tunnel.L: Average number of jobs in the system ģ 3 Time in System (W) 2 2 1 1 1 2 3 Packet # (N) Proof: Method 1: Definition # in System (L) = 1 2 3 4 5 6 7 8 Time (T).Applies to any system in equilibrium, as long as nothing in black box is creating or destroying tasks System Arrivals Departuresģ 3 3 # in System 2 2 2 1 1 1 1 2 3 4 5 6 7 8 Time Time in System 1 2 3 Packet # Proving Little’s Law Arrivals Packet # Departures 1 2 3 4 5 6 7 8 Time J = Shaded area = 9 Same in all cases!.Observed before, Little was first to prove.Little’s Law: Mean number tasks in system = arrival rate x mean response time.Probability queue is at a certain length.Queuing theory provides probabilistic analysis of these queues.Queuing Theoryand Traffic Analysis CS 552 Richard Martin Rutgers University