Queuing Networks Modeling Virtual Laboratory

Dr. S. Dharmaraja Department of Mathematics

IIT Delhi http://web.iitd.ac.in/~dharmar

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Outline

• Introduction • Simple Queues • Performance Measures • Performance Measures • Case Study

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Queuing System

Customer Arrivals

Queue (waiting line)

Customer Departures

Arrivals (waiting line)

Server Departures

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Kendall Notation: A/B/C/D/X/Y

A: Distribution of inter arrival times B: Distribution of service times C: Number of servers D: Maximum number of customers in D: Maximum number of customers in system X: Population Y: Service policy

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Queuing Models

• M/M/1/∞/FCFS, M/Ek/c/∞/LCFS, M/D/c/k/FCFS, GI/M/c/c/FCFS, MMPP/PH/1/FCFS are some of the examples of the queuing models.

• The notation M/M/1/∞/FCFS indicates a queuing process • The notation M/M/1/∞/FCFS indicates a queuing process with the following characteristics: § Exponential Inter Arrival Times, § Exponential Service Times § One Server § Infinite System Capacity, § First Come First Served Scheduling Discipline.

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Some Basic Relations

• Cn: nth customer, n=1,2,… • An: arrival time of Cn

• Dn: departure time of Cn • Dn: departure time of Cn

• α(t): no. of arrivals by time t • δ(t): no. of departures by time t • N(t): no. of customers in system at time t,

N(t) = α(t) - δ(t).

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Diagrammatic Representation

α(t)=5, d(t)=2

No. of customers

N(t)=3 customers

1

a1 a2 d1 a3 a4 a5 t d2 Queues Notes 1

Time

d3 7

The M/M/1 Queue

• Arrival process: Poisson with rate λ • Service times: iid, exponential with parameter µ • Service times and interarrival times: independent • Single server • Single server • Infinite waiting room • N(t): Number of customers in system at time t (state)

λ λ λ λ

0 1 2 n n+1

µ µ µ µ

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M/M/1 Queue: Markov Chain Formulation

• Transitions due to arrival or departure of customers • Only nearest neighbors transitions are allowed. • State of the process at time t: N(t) = i ( i ≥ 0). • {N(t): t ≥ 0} is a continuous-time Markov chain with • {N(t): t ≥ 0} is a continuous-time Markov chain with

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M/M/1 Queue: Stationary Distribution

• Birth-death process µp n =λp ⇒n−1

λ n p n =

µ p =ρp = =ρ p n−1 n−1 0 µ

• Normalization constant∞

p =1⇔ p ⎡1+

∞n

ρ⎤

=1⇔ p =1− ρ , if ρ <1 ∑n=0

n 0 ⎢⎣∑n=1

⎥⎦ 0

• Stationary distribution

p =ρ n (1−ρ), n= 0,1,... n

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Performance Measures

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Performance Measures

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Blocking Probability

• An important design criterion is the blocking probability of a queuing system • Would we be happy to lose one packet in 100?, 1000? 1000,000? 1000? 1000,000? • How much extra buffer space must we put in to achieve these figures?

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Blocking Probability

• If a queue is full when a packet arrives, it will be discarded, or “blocked” • So the probability that a packet is blocked is

exactly the same as the probability that the queue exactly the same as the probability that the queue is full

• That is, PB = pN

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Blocking Probability

• Schwartz has this useful diagram to describe throughput and blocking

λ = load

Queue γ = throughput

λPB

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= λ(1-PB)

15

M/M/c Queue λ λ λ λ

0 1 2 c c+1

µ 2µ cµ cµ

• Poisson arrivals with rate λ Exponential service times with parameter µ • Exponential service times with parameter µ

• cservers • Arriving customer finds n customers in system

- n < c: it is routed to any idle server - n ≥ c: it joins the waiting queue - all servers are busy

• Birth-death process with state-dependent death rates

µ n ⎧nµ,

=⎨⎩cµ,

1≤ n≤c

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M/M/c Queue

• Steady state solutions

• Normalizing

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Performance Measures

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Performance Measures

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M/M/c/K Queues

• Poisson arrivals with rate λ • Exponential service times with parameter µ • cservers with system capacity K • Arriving customer finds n customers in system • Arriving customer finds n customers in system

- n < c: it is routed to any idle server - n ≥ c: it joins the waiting queue - all servers are busy

• Customers forced to leave the system if already K present in the system.

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M/M/c/K Queues

• Birth death process with state dependent death rates

• Stead

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Performance Measures

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Performance Measures

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M/M/c/c Queue λ λ λ

0 1 2 c

µ 2µ cµ

� c servers, no waiting room� c servers, no waiting room� An arriving customer that finds all servers busy is blocked� Stationary distribution:

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Case Study 1

Cellular Networks

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Components of Cellular Systems

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Wireless Handoff Performance Model

Common Channel Pool

New Calls (NC) Call completion

Handoff Calls (HC) From

neighboring cells

A Cell

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Handoff out To neighboring cells

27

System Description

• Handoff Phenomenon: - A call in progress handed over to another cell due to user mobility - Channel in old base station is released and idle channel given in

new base station § Dropping Probability § Dropping Probability

§ If No idle channel available, the handoff call is dropped. § Percentage of calls forcefully terminated while handoff.

§ Blocking Probability § If number of idle channels less than or equal to `g’, new call is

dropped § Percentage of new calls rejected.

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Basic Model

• Calls arrives in Poisson manner. - λ1:Rate of Poisson arrivals for NC

- λ2:Rate of Poisson arrivals for HC

• Exponential service times - µ1:Rate of ongoing service - µ2:Rate of handoff of call to neighboring cell.

• N: Total number of channels in pool • g: No. of guard channels.

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Basic Model

• C(t): No. of busy channels, • {C(t), t ≥0}: continuous time, discrete state Markov

process,• Only nearest neighbor transitions allowed• Only nearest neighbor transitions allowed• Variation of M/M/c/c queuing model.

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State Transition Diagram

λ

0 1

λ

….

λ λ

…. n

λ λ N-g-1 N-g

λ2 λ2

… N 0 1 µ1 µ2

n

µn µn+1 µN-g-1 µN-g µN-g-1 µN

Markov Chain Model of Wireless Handoff

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Steady State Equations

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Steady State Solutions

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Loss Probabilities

• Dropping Probabilities

• Blocking Probabilities

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Text and Reference Books

• Main Text Books - Data Networks, Dimitri P. Bersekas and Gallager, Prentice Hall, 2nd edition, 1992 (chapters 3 and 4). - Probability and Statistics with Reliability, Queuing

and Computer Science Applications, Kishor S. Trivedi, and Computer Science Applications, Kishor S. Trivedi, John Wiley, second edition, 2001 (chapters 8 and 9).

• Reference Books - The Art of Computer Systems Performance Analysis: Techniques for Experimental Design, Measurement, Simulation, and Modeling, Raj K. Jain, Wiley, 1991. - Computer Applications, Volume 2, Queuing Systems, Leonard Kleinrock, Wiley-Interscience, 1976.

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