Integral Assignment QEES
Part 3: Packet losses modelling and evaluation with Markov Chains
Anne Remke and Björn F. Postema
University of Twente
Design and Analysis of Communication Systems
[email protected]
January 6, 2014
Exercise 1
Wireless communication inevitably introduces time delays and message losses, which may degrade
the system reliability and network performance. In this part of the integral assignment, you
will use Discrete-time Markov Chains (DTMCs) to calculate the effect of packet losses on the
network performance.
As seen in the previous exercise, the WirelessHART protocol uses the TDMA protocol for the
communication between nodes. In this part of the assignment we consider the WirelessHART
network in the figure below.
Figure 1: Connectivity graph of a typical WirelessHART network
1
Link model
Consider a dynamic link where the received signal strength is above an acceptable threshold
part of the time, and below the threshold when the line of sight is blocked. This can be modeled
as a DTMC with two states, namely UP and DOWN, as shown in Figure 2.
In the UP state, the transmission error probability is negligible, however in the DOWN state,
the received signal strength is so low that the error probability is very high. In case the link is
UP, the entire message will be transmitted successfully without any bit error; in case the link is
DOWN, the message transmission fails due to one or more bit errors and the message needs to
be re-send later. The state of the link remains unchanged during one slot and may change in
the next slot. The probability that the signal strength becomes too low in the next slot is pf l
and the probability that a link recovers is denoted prc .
pf l
1-pf l
UP
DOWN
1-prc
prc
Figure 2: Two-state DTMC link model
WirelessHART radio used the modulation technology OQPSK (Offset quadrature phase-shift
keying). The Bit Error Rate of OQPSK modulation in a AWGN (Additive white Gaussian noise)
channel is given by:
r !
1
Eb
BEROQP SK = erf c
,
(1.1)
2
E0
1
where erf c() represents the complementary error function, and Eb /E0 represents the energy per
bit to noise power spectral density ratio, which is a normalized Signal-to-Noise Ratio (SNR)
measure and can be regarded as the ”SNR per bit”. The received SNR can be measured using
pilot packages that are transmitted from one node to the other via the wireless link.
The successful transmission of each bit (with probability 1 − BER) then follows a Bernoulli
distribution, hence, assuming the typical WirelessHART message is L bits long, the failure
probability is given by:
pf l = 1 − (1 − BER)L .
(1.2)
In a WirelessHART network, a message is transmitted successfully if and only if the wireless
link remains operational in that slot. Therefore, in the following we assume that the steady
state probability ps of the link to be up determines the probability that a message is correctly
transmitted on a given link and the steady state probability to be down pf represents the
probability that a message is lost on a given link.
2
Path model
Using a hierarchical modeling approach, we can then model how the information that is generated
at the sensor is forwarded through the network, taking into account the probability that a certain
link is operational.
Consider the following three-hop path n1 → n2 → n3 → G as an example. The communication schedule η = (∗, hn1 , n2 i, ∗, ∗, hn2 , n3 i, ∗, hn3 , Gi) specifies that the first transmission which
is relevant for this path takes place in slot 2. Slots with a ∗ are not relvant for the path under
investigation but are used for the communication of other paths in the same network. The frame
size is then given by Fs = 7 and in the following we assume that the reporting interval equals
one, that is fresh information is produced in every cycle of 7 slots and is discarded if it does not
reach its destination within one cycle.
R7
ps3
6,6,6
1
7,7,7
ps2
ps1
1,-,-
1
2,-,-
3,3,-
pf 1 3,-,-
1
4,4,-
1
1 4,-,-
1
pf 3
5,5,- pf 2 6,6,5,-,-
6,-,-
1 7,7,1 7,-,-
1
1
Discard
Figure 3: DTMC diagram of the path model of a three-hop path when Is = 1
In the following, we construct a DTMC that models the age of the information at every
hop of the path. For a path with n hops, each state is denoted with a label (x1 , x2 , . . . , xn ),
where xi represents the age of the message at hop i. In the initial state is (1, −, −), the sensor
has just produced fresh information and hence the age of the information at the sensor is 1.
Then, depending on the communication schedule, in each slot either a transmission takes place
or the slot passes, since it is used for a different path. In the schedule η, the first slot is idle,
so the DTMC moves to the second state (2, −, −) with probability 1. In the second slot, the
communication schedule entry hn1 , n2 i indicates a transmission on the link between n1 and n2 .
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Then the DTMC moves from state (2, −, −) to state (3, 3, −) with probability ps (modeling a
successful transmission), and to state (3, −, −) with probability pf (modeling a transmission
failure).
The path DTMC is constructed following the rules in [1] and shown in Figure 3. Note that
after Fup = 7 steps, either the goal state R7 is reached the goal state at the seventh slot or the
‘Discard’ state is reached when the Time To Live (TTL) of the message has reached zero at the
end of this cycle.
3
Reachability
In the following R denotes the probability that a message generated at the source node reaches
the gateway before the end of a given reporting interval (reachability). If a message fails to
reach the gateway, then the input signal I is lost, possibly causing instability of the control loop.
Probability R is then given by the sum of the transient probabilities of the goal states at
the end of the reporting interval. Note that in case of a reporting interval of 1 the DTMC has
a single goal state G1 (corresponds to R7 in Figure 1), in case of a higher reporting interval
Is = y, the DTMC has y goal states G1 , . . . , Gy .
R=
Is
X
pGi (t) for t = Is ∗ Fs .
(3.1)
i=1
4
Delay
In WirelessHART networks excessive delay can lead to a significant degradation in system
performance. Delay is defined as the time difference between the born time Tborn and the
reception time Trec , which equals the age of a message in the path model. The delay distribution
τ can also be derived from the transient distribution of the DTMC model.
The age measured in slots has to be converted to the absolute time in millisecond. For each
delay di , the delay probability is the percentage of messages with delay di among all the received
messages, i.e. the averaged transient probability. This is given by:
pG (t)
pG (t)
= i
for t = Is ∗ Fup .
τ (di ) = PIs i
R
p
(t)
j
j=1
(4.1)
Therefore, the expected delay E[τ ] is defined as
E[τ ] =
Is
X
di ∗ τ (di ).
(4.2)
i=1
5
Questions
Describe your steps carefully. Models and properties should be attached as files and they
should be in your report. And do not forget to make use of supporting material, namely: The
WirelessHART paper [1], Prism manual [2] and Prism tutorial [3].
1. Calculate the success probabilities ps and the failure probabilities pf for links that are in
steady-state during transmission, using an energy per bit to noise power spectral density
ratio of 6.5514, a WirelessHART message length of 1016 bits and a recovery probability
prc = 0.8. Indicate the probability matrix of the link DTMC and show the vector matrix
multiplications that leads to the steady-state probabilities.
2. (a) Create a path model DTMC in PRISM for the three-hop path from S1 → S2 →
Gateway → A1 using the following parameters:
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• the optimal communication schedule is
(hS1, S2i, hS3, Gatewayi, hS2, Gatewayi, hGateway, A1i, hGateway, A2i),
• the failure and success probabilities as obtained in the previous answer,
(Hint: Use constants for pf and ps )
• and a reporting interval of Is = 1 and the super-frame size Fs = 5.
(b) Specify the following properties in PCTL using PRISM notation:
i. What is the probability for a packet in this model to be discarded?
ii. What is the probability to reach a goal state before time t?
(c) Calculate the expected delay for this scenario. Note that you probably have to do the
final computations outside PRISM.
(d) Create a graph that shows the probability of reaching a goal state before each time
point t ∈ {0, 1, . . . , Is ∗ Fs }. (Hint: Create an experiment with ’time’ as a constant.)
3. Assume that the reporting interval is changed from Is = 1 to Is = 2, so now there are two
goal states.
(a) Create a graph that shows the probability of reaching a goal state before each time
point t ∈ {0, 1, . . . , Is ∗ Fs }.
(b) How is this graph different from the one computed in 2(d)? Explain where the differences
come from.
(c) What is the expected delay?
4. Assume that the recovery probability is improved from 0.8 to 0.99 for your model from
Question 3.
(a) Create a graph that shows the probability of reaching a goal state before each time
point t ∈ {0, 1, . . . , Is ∗ Fs }.
(b) What is the expected delay?
(c) How do the reachability and the expected delay change compared to the model with a
recovery probability of 0.8? Explain the differences.
5. You have now modeled three different aspects of the same system with three different
modeling formalisms. Explain why the specific formalism has been chosen for each aspect.
Reflect on your experiences using the different tool sets.
For more insight and examples read the following:
[1] Remke, A., & Wu, X. (2013). WirelessHART modeling and performance evaluation. 2013
43rd Annual IEEE/IFIP Int. Conf. on Dependable Systems and Networks (DSN), 112.
doi:10.1109/DSN.2013.6575358
[2] Prism. Prism Manual. http://www.prismmodelchecker.org/manual/Main/AllOnOnePage
[3] Prism. Prism Tutorial. http://www.prismmodelchecker.org/tutorial/
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