Gradual Delay Differentiation in Priority Scheduling

1
Gradual Delay Differentiation in Priority
Scheduling

• Tom Maertens, Joris Walraevens and Herwig Bruneel
• Ghent University (UGent)
• Department of Telecommunications and Information
  Processing (TELIN)
• Stochastic Modelling and Analysis of
  Communication Systems (SMACS)
• Sint-Pietersnieuwstraat 41, B-9000 Ghent, Belgium

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Framework

• Modern telecommunication networks are designed to
  offer a wide variety of services
• information access
• e-mail
• internet telephony
• file sharing
• streaming media
• Different services have extremely diverse
  Quality-of-Service (QoS) requirements
• real-time services do not tolerate delay
• non-real-time services are quite vulnerable to
  loss and require a large throughput

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Service differentiation

• The traffic that flows through telecommunication
  devices nowadays can thus more or less be
  classified into two types
• Delay as QoS-measure delay-sensitive (type-1)
  traffic versus delay-tolerant (type-2) traffic
• To achieve the required delay differentiation
  between both types of traffic, delay-sensitive
  traffic is prioritised in scheduling packets for
  transmission

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Assumptions

• Physical structure of the system
• discrete time
• infinite storage capacity, divided in two
  priority queues
• one transmission channel
• Transmission process
• work-conserving
• single-slot transmission times
• transmissions are synchronised to the slot
  boundaries

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Static priority scheduling

• Priority is always given to delay-sensitive
  packets
• delay-tolerant packets can only be transmitted
  when there are no delay-sensitive packets present
  in the system
• priority levels of both types of traffic never
  change during time

time
6
Performance of the static priority scheduling
discipline

• Low delays for delay-sensitive packets
• Possibly excessive delays for delay-tolerant
  packets, especially when the system is highly
  loaded and much network traffic is
  delay-sensitive

Gr!_at_
time
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Dynamic priority scheduling

• Priority levels of the two types of traffic can
  change during time, so delay-tolerant packets can
  also be transmitted when there are
  delay-sensitive packets present in the system
• Dynamic priority scheduling disciplines aim for a
  more gradual delay differentiation between both
  types of traffic
• Two categories
• varying priority levels
• priority jumps the priority level of
  delay-tolerant packets can increase in the course
  of time

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Priority jumps

• Packets of the low-priority queue can jump to the
  high-priority queue in the course of time
• Jumped delay-tolerant packets are treated in the
  high-priority queue as if they are
  delay-sensitive packets
• Jumps occur at the end of slots

time
9
Jumping criteria to decide if and when
delay-tolerant packets jump

• A maximum queueing delay in the low-priority
  queue
• A queue-length-threshold for one of the queues
• A random jumping probability per time unit
• An arrival characteristic of one type of traffic

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Jumping mechanisms

• Merging the high- and low-priority queues in a
  slot
• every slot Merge-Every-Slot (MES)
• with a certain probability Merge-By-Probability
  (MBP)
• Letting only one packet jump in a slot
• always Jump-Or-Transmit (JOT)
• with a certain probability Jump-By-Probability
  (JBP)
• Jump in a slot also depends on the number of
  arrivals
• delay-sensitive (type-1) packets
  Jump-If-Arrivals-of-type-1 (JIA1)
• delay-tolerant (type-2) packets
  Jump-If-Arrivals-of-type-2 (JIA2)

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Example Merge-By-Probability

• At the end of a slot, the total content of the
  low-priority queue jumps with probability ? to
  the end of the high-priority queue (i.e., both
  queues are merged with probability ?)

time
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Arrival process

• Two types of packets
• type-1 delay-sensitive
• type-2 delay-tolerant
• Number of arrivals of both types of packets
  (denoted by a1 and a2 respectively)
• are independent and identically distributed
  (i.i.d.) from slot to slot
• can be correlated within one slot

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Determining the system equations

• Describe the evolution of the queue contents from
  slot to
• slot
• uH,k content of the high-priority queue at the
  beginning of slot k
• uL,k content of the low-priority queue at the
  beginning of slot k

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Transforming to a functional equation

• Introducing probability generating functions
• Assuming that the system evolves towards a steady
  state ! dropping the time index k

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Solving the functional equation

• Determining the constant U(0,0) and the functions
• U(0,z2) en U(z1,z1) leads to the joint
  probability
• generating function of the contents of both
  queues

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Delay of a type-1 packet

• Jumps always occur at the end of a slot ! all
  type-1 packets that arrive during a slot enter
  the high-priority queue in front of jumping
  type-2 packets
• Is only determined by the content of the
  high-priority queue at the moment of arrival

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Delay of a type-2 packet

• Priority scheduling new type-1 packets can
  arrive while type-2 packets are waiting in the
  low-priority queue, and these type-1 packets have
  priority
• Jumping mechanism type-2 packets can jump to the
  high-priority queue in the course of time
• Combination of these two characteristics of the
  model makes the analysis not always
  straightforward!

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Results

• Probability generating functions of
• the contents of the two priority queues
• the delays of both types of packets (for most
  models)
• Performance measures
• moments via the moment generating property of
  probability generating functions
• approximate tail distributions via the
  dominant-singularity method applied on
  probability generating functions ? not
  necessarily exponentially decaying tail
  probabilities

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Numerical examples packet switch

• Arrival at an input port
• occurs with probability ?T ( arrival rate)
• and is of type 1 with probability ? ( fraction
  of type-1 arrivals in the overall traffic mix)
• Uniform and independent routing towards the
  output ports

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Mean delays of both types of traffic for ?T0.9
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Mean delays of both types of traffic for ?T0.9
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Conclusions

• Priority schemes with priority jumps build upon
  the simplicity and efficiency of the static
  priority scheme, but prevents delay-sensitive
  traffic from starving
• Depending on the delay requirements of the
  different types of traffic, we can introduce and
  tailor one of the jumping mechanisms
• Analysis based on probability generating
  functions
• can overcome mathematical challenges (e.g., the
  calculation of boundary functions)
• is useful for the calculation of different
  performance measures (such as moments and tail
  probabilities)