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1 Chapter 8 Queueing models

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2 Delay and Queueing Main source of delay Transmission (e.g., n/R) Propagation (e.g., d/c) Retransmission (e.g., in ARQ) Processing (e.g., running time of protocols) Queueing Queueing Theory Study of mathematical queueing models Since early 1900’s by Erlang

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3 Queue Customer System Service Delay box : Multiplexer switch network Message, packet, cell arrivals Message, packet, cell departures T seconds Lost or blocked

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4 Queueing Discipline One customer at a time First-in first-out (FIFO) LIFO Round robin Priorities Multiple customers at a time FIFO Separate queues/separate servers Blocking rule Discard when full Drop randomly Block a certain class

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5 Definitions T: Time spent in the system A(t): # of arrivals in [0,t] B(t): # of blocked customers in [0,t] D(t): # of departures in [0,t] N(t): # of customers in the system at t N(t)=A(t)-D(t)-B(t) Long term arrival rate Throughput Average number in the system Fraction of blocked customers

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6 A(t) t n-1 n n+1 Time of nth arrival = 1 + n Arrival Rate n arrivals 1 + n seconds = 1 = 1 ( 1 + n )/n E[]E[] 11 22 33 nn n+1 Arrivals

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7 Little’s Law If the system does not block customers, then E{N} = λ E{T} If the block rate is Pb, then E{N} = (1-Pb) λ E{T} Proof

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8 Arrivals and Departures A(t) D(t) T1T1 T2T2 T3T3 T4T4 T5T5 T6T6 T7T7 Assumes first-in first-out C1C1 C2C2 C3C3 C4C4 C5C5 C6C6 C7C7 C1C1 C2C2 C3C3 C4C4 C5C5 C6C6 C7C7 Arrivals Departures

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9 Examples Let the arrival rate be 100 packets/sec. If 10 packets are found in the queue in average, then the average delay is 10/100=0.1 sec. Traffic is bad in a rainy day. For the same volume of customers, a fast food restaurant requires smaller dining area.

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10 Example E{N} = λ E{T}

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11 Basic Queueing Models 1 2 c A(t) t D(t) t B(t) Queue Servers Arrival process X Service time ii i+1

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12 Arrival Processes Interarrival times 1, 2,… Arrival rate =1/E{ } Statistics Deterministic Exponential interarrival times Poisson arrival process

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14 Service Processes Service times X 1, X 2,… Processing capacity =1/E{X}

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15 Service times X M = exponential D = deterministic G = general Service Rate: = 1/E[X] Arrival Process / Service Time / # of Servers / Max Occupancy Interarrival times M = exponential D = deterministic G = general Arrival Rate: = 1/E[ 1 server c servers infinite K customers unspecified if unlimited Multiplexer Models: M/M/1/K, M/M/1, M/G/1, M/D/1 Trunking Models: M/M/c/c, M/G/c/c User Activity: M/M/ , M/G/ Queueing System Classification

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c X N q (t) N s (t) N(t) = N q (t) + N s (t) T = W + X W P b (1 – P b ) N(t) = number in system N q (t) = number in queue N s (t) = number in service T = total delay W = waiting time X = service time Queueing System Variables E{N} E{N q } E{N s } Traffic load Utilization

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17 Poisson arrivals rate K – 1 buffer Exponential service time with rate The M/M/1/K Model Average packet transmission time E{X} = E{L}/R Maximum service rate =R/E{L} P{1 arrival in Δt} = Δt + o(Δt)

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n-1 n n ( t 1 - t t t M/M/1 Steady State Probabilities Average number of customers in the system? Average delay? Average wait time?

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19 m separate systems versus One consolidated system m mm Example: Effect of Scale What is the average delay of the combined system?

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20 Poisson arrivals rate Infinite buffer = 1/E[X] The M/G/1 Model The average wait time: Similar to M/M/1 except that the service time X may not be exponentially distributed.

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21 Proof

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22 Blocked calls are cleared from the system; no waiting allowed. Performance parameter: P b = fraction of arrivals that are blocked P b = P[N(t)=c] = B(c,a) where a= The Erlang B formula, valid for any service time distribution B(c,a) a c c! a j j! j=0 c Many lines Limited number of trunks 1 2 c N(t ) PbPb = (1–P b ) Poisson arrivals E[X]=1/ Erlang B Formula: M/M/c/c

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c-1 c 1 - ( t1 - ( t 1 - ( c-1) t1 - c t1 - t t t tc t State Transition Diagram

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