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Real-Time Queueing Network Theory

Presented by Akramul Azim

Department of Electrical and Computer EngineeringUniversity of Waterloo, Canada

John P. Lehoczky

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Outline

• Fundamentals

– Queueing theory– RT queueing theory

• Contributions

– RT network queueing theory– Analysis– Simulation studies– Reducing lateness

• Critics and Reviews

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Queueing Theory - Basics

• Queuing Theory deals with systems of the following

type:

InputProcess

OutputProcess

Queue

Example Input Process Output Process

Bank Customers arrive at

bank

Tellers serve the

customers

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Queueing Theory - Terminology and Notation

• State of the system

Number of customers in the queueing system (includes customers

in service)

• Queue length

Number of customers waiting for service

= State of the system - number of customers being served

• M/M/1 Queue

Single Server/Queue, arrivals follow poisson process, and job

service times have an exponential distribution

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Queueing Theory - Terminology and Notation

• N(t) = State of the system at time t, t ≥ 0

• Pn(t) = Probability that exactly n customers are in the queueing system at time t

• n = Mean arrival rate (expected # arrivals per unit time) of new customers when n customers are in the system

• s = Number of servers/queues (parallel service channels)

• n = Mean service rate for overall system (expected # customers completing service per unit

time) when n customers are in the system

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Real-Time Queueing Theory

• Applications/tasks/jobs/packets have explicit timing requirements (deadlines)

• Stochastic task arrivals, execution time, and network routing

• Avoid over-provisioning of resources

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Real-Time Queueing Network Theory

• Real-time network queueing theory combines,

1) Hard real-time system scheduling theory

2) Heavy traffic queueing theory

3) Jackson network theory

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Hard Real-time System Scheduling Theory

• Typical hard real-time system requirements

• Scheduling policies like EDF

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Heavy Traffic Queueing Theory

• Heavy traffic is the case when traffic intensity nears 1

• Timing behaviour of a RT queue becomes nearly deterministic in heavy traffic

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Jackson Network

• A Jackson network is connects queue (e.g., M/M/1) at each node

• Tasks may arrive any node in the network

• Tasks have end-to-end deadlines

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RT Queueing Theory Terminologies

• Lead-time (l): The time remaining until the task’s deadline.

• Lateness: Negative lead-times

• State variable: (N,l1,…,lN), where N is the number of tasks

• Lead time vector: (l1(t),…,lQ(t(t)), where Q(t) is the number in the system at time t

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RT Queueing Network Terminologies

• P = (pij): K ×K routing matrix where pjk represents the probability

that a task leaving node j goes to node k

• = (1,…, K), the arrival rates for independent Poisson external

task arrival process to each node

• Γ = (Г1 ,…, Гk), initial deadline distribution

• μ= (μ1, …, μk) , the service rates

• θ= (θ1, …, θk), the total arrival rate to each node

• ρj = θj/ μj , the traffic intensity of total task arrivals

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Facts – Queueing Theory

• Lead-time vector depends on three factors:

1. The number of tasks in the queue

2. The tasks deadline distribution

3. The queue discipline (scheduling policy)

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Facts – Queueing Network Theory

• Lead-time vector of a node depends on four factors:

1. The total task arrival rate

2. The tasks deadline distribution of that node

3. The queue discipline (scheduling policy)

4. the occupancy of the node (queue length)

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Experimental Analysis

• Simulation study performed

• Assume 2 stations in the network

• Nodes have a Queue, λ= .95, traffic intensity = 0.95

• Task deadlines are at least 40

• Two clocks are used: (1) Global (2) Local

• Global clock advances constantly

• Local clock advances only when queue lengths satisfy

some precondition

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Queue 1: Deadlines = 60 + 30 exp(1), Local Clk= 20

Note: 60 + 30 exp(1) = 141.54

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Queue 1: Deadlines = 60 + 30 exp(1), , Local Clk= 40

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Queue 1: Deadlines = 60 + 30 exp(1), , Local Clk= 60

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Queue 2: Deadlines = 60 + 30 exp(1), , Local Clk= 20

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Queue 2: Deadlines = 60 + 30 exp(1), , Local Clk= 40

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Queue 2: Deadlines = 60 + 30 exp(1), , Local Clk= 60

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Queue 1: Deadlines = 30 + 60 exp(1), , Local Clk= 20

Note: 30 + 30 exp(1) = 193.09

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Queue 1: Deadlines = 30 + 60 exp(1), , Local Clk= 40

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Queue 1: Deadlines = 30 + 60 exp(1), , Local Clk= 60

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Queue 1: Deadlines = 30 + 60 exp(1), , Local Clk= 80

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Queue 2: Deadlines = 30 + 60 exp(1), , Local Clk= 20

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Queue 2: Deadlines = 30 + 60 exp(1), , Local Clk= 40

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Queue 2: Deadlines = 30 + 60 exp(1), , Local Clk= 60

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Queue 2: Deadlines = 30 + 60 exp(1), , Local Clk= 80

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Simulation Findings

• Queueing theory can analyze complex network

structures

• Task lead time profiles with reasonable accuracy at high,

but not extreme

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Facts - Lateness

• If task deadlines are 40, approximately 40% of the tasks

will miss their deadlines

• If task deadlines are increased to 60, still 20% will be

late.

• This holds even though the best scheduling policy used

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Control Policies to Reduce Lateness

• Blocking policy:

1. Tasks are blocked if the total number of tasks

reaches some level

2. Individual queue reaches some threshold

• Abort policy:

1. Tasks should be aborted as soon as they become late

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Blocking Probabilities (Mean task deadlines = 40)

Blocking strategy can reduce the task lateness from 40% to 4%

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Critics

• Unable to provide guarantee of meeting deadlines of all tasks in the worst-case

• Simulation study performed with distributions but not any real data set or application

• Worst-case resource constraints not taken into account

• Analysis on bounding buffer requirements is absent

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Conclusions and Future Work

• Can be used for modelling probabilistic real-time systems

• Can be used to assess queue control policies

• Can analyze large-scale soft-RT systems/networks

• Future work may include periodic arrival process and deriving

bounds on queue length or buffer requirements.

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Thank You. Any Questions?

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