In this paper, the authors make model and evaluate the duty cycle management algorithm using the theory of discrete time Markov chains and M/G/1/K queues with vacations. The network uses two types of channel access mechanism, one based on slotted CSMA-CA algorithmin which the slots are aligned with the beacon frames sent periodically by the PAN coordinator, and another based on unslotted CSMA-CA similar to IEEE 802.11, in which case there are no beacon frames. The most important requirements for sendor networks is the maximation of their lifetime due to high costs of maintenance activities. Sensor lifetime can be extended by adjusting the frequency and ratio of active and inactive periods of sensor nodes.

In this paper, there are so many mathematical equations🙂

**The Sensor Utilization parameters and terms:**

The probability distribution of the packet service time in the MAC sublayer described by the Probability Generating Function (PGF), The Laplce-Stieljes Transform (LTS), the probability distribution function of the packet service time, ptobability density function, mean duration of the packet service time, the probability to access the medium, the probabilities that the medium is idle on the first and second CCA, network buffer, the packet arrival process to sensor i, poisson process with arrival rate, FCFS and exhaustive fashion, the duration of vacation period, geometrically distributed, mean duration of the vacation, the probabilities of k packet arrivals to the sensor buffer during the packet service time and during the sleep time, PGF for the number of packet arrivals to the sensor buffer during the packet service time and sleep time, the steady state probabilities that there are k packets in the device buffer immediately upon the departure of a packet and after returning from vacation, the steady state for state transitions, the probability distribution of the device queue length at the time of packet departure, the probability that the vacation follows the artitrary Markov point, the probability that the packet service follows the arbitrary Markov point, the mean distance between two Markov point, the probability that the sensor is active, offered load to the system, the probability that the packet will not be admitted to the sensor buffer due to insufficient space, the mean number of active sensors, the sensor realibility per node, the number of packets that each sensor delivers to the coordinator per time unit, the time between the start of two consecutive vacations (vacation cycle), the duration of busy period, the number od packets sent during the busy period, the mean period between two packet transmission, the number of packets received by the coordinator, the duration of basic backoff period from MAC algorithm, the total length of physical headers and MAC headers and trailer, the mean packet length at the MAC sublayer, the throughput, network reliability, the packet service time in the MAC sublayer, the packet arrival rate, packet length and the number of nodes.

**Analysis of The MAC Sublayer :**

The time needed to transmit a packet from the head of the queue (includes the time from the moment when the CSMA-CA algorithm has started to the moment when the receipt of the packet has been acknowledged by the destination device), the probability that the system enters the vacation state which is the probability that the device buffer is empty upon packet departure, the probability to stay in the vacation state, the value of the backoff time counter, the value of NB at time t, the value of CW at time t, the current value of the “delay line counter” which is started if the transmission can’t be finished within the current superframe, fixed packet size (including MAC and PHY headers), the period between the packet and its acknowledgement, the duration of the acknoledgement packet, the number of backoff periods necessary to complete the transmission within the current superframe,the probability that the remaining time within the superframe will not suffice to complete the transmission, superframe size, the idle state with no packets to transmit, the constant macMaxCSMABackoffs which represents the maximum value of the variable NB, the current value of NB during the execution of the algorithm, the maximum value of the random waiting time, the non-null transition probabilities, the stationary distribution of the chain, All probabilities in the Markov chain, the total probability to access the medium, the probability to access the medium when the transmission is deferred to the next superframe due to lack space and to access the medium in the current superframe, the probabilities that medium is idle on the first and second CCA, the number of active stations follows the binomial distribution, the number of active stations, the number of non-delayed and delayed packet transmission, a binomial distribution, the probability that one or more delayed packets will be transmitted in one superframe will take place,the number of busy backoff periods due to these transmission, the mean period between two transmissions of non-delayed packets, the mean number of non-delayed packet transmission within the superframe, the factor that the first two CCA, the probability that the medium is idle at the first CCA, the probability that medium is idle at the second CCA, the probability of a transmission going on.

**The Probability Distribution For the Packet Service Time :**

The PGF of the packet length, the mean packet size, the PGF of the interval between the end of the packet and the arrival of the corresponding acknowledgement (macMaxAckWait), the PGF of the acknowledgement duration, the PGF of the duration of the beacon frame, macBeaconOrder, the probability distribution of the packet service time at the MAC sublayer, the service time for the packet queue at the network node, the probability that a backoff period is the last one within the active superframe, the PGF for the effective duration of the backoff period (including the duration of the beacon frame), the probability that the remaining number of backoff periods will not suffice for the transmission, the number of backoff periods needed for an acknowledged transmission, the PGF for the number of backoff periods that are waste due to the sufficient space in the current superframe, the PGF of the packet transmission time, the PGF of the packet service time, the PGF for the packet service time, the first two moments of

**Traffic Performance in PAN assumption:
**raw data rate 250 kbps and SO,BO=0, aUnitBackoffPeriod = 10 bytes, aBaseSlotDuration = 30 bytes, aNumSuperframeSlots = 16, aBaseSuperframeDuration = 480 bytes, macMinBE = 3, aMaxBE = 5, macMaxCSMABackoffs = 4, the physical layer header = 6 bytes, the MAC sublayer header and Frame Check Sequence field = 9 bytes, the number of sensor nodes = 10 – 100.

Very difficult to understand🙂

**Source :**

Queueing Analysis of Sleep Management in an 802.15.4 Beacon Enabled PAN.

By Jelena Misic and Vojislav B Misic

Note : This resume is created for self-learning only. Author and Publisher hold copyrights

*Think Out The Box
*

*October 18, 2008*

*Taipei City
High Speed Network Lab*

Udin Harun