Real-Time Queueing Network Theory
John P. Lehoczky
Presented by
Akramul Azim
Department of Electrical and Computer Engineering
University of Waterloo, Canada
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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:
Input
Process
Output
Process
Queue
Example
Bank
Input Process
Output Process
Customers arrive at
Tellers serve the
bank
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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