Handover model with MCP

A Markov chain is a stochastic model describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event [148]. The basic property of this stochastic process is 'memorylessness', which means we can predict the situation of the system in the future according to current situation only, even without knowing the history of the whole process. MCP has been widely applied in establishing the mathematical model for real-world process, which contains random objects and time factors, such as predicting the customers arriving the specific item’s arrival sequence, the price of stock and growth rate of observing species [149] – [150].

After analysing how MCP works, we will be able to model Handover process with MCP. In introduction section, we have stated that a complete Handover contains two phases: initialization and process. Take both Handover from macro to small and from small to macro, there are total four main states: M, S, I, I'. We define M states representing UE is bound to

macro cell and undergoes initialization phase. Similarly, S states representing UE is bound to small cell and undergoes initialization phase. Besides that, we define I states as handover process from macro to small cell, and I’ states as handover process from small to macro cell. During initialize phase, Time-to-Trigger (TTT) is the crucible parameter, which restricts UE from entering second phase unless the candidate’s cell's RSS plus virtual bias is larger than current cell's RSS for a period of pre-defined time. In order to map TTT into our MCP model, we divide the whole TTT into several Transmission Time Interval (TTI), and each TTI represents a state within S = {S1, S2, ...Sn} or M = {M1, M2, ...Mn}. The conventional TTT

time may vary from 40ms to 100ms and conventional TTI time is 10ms [152]. As a result, we have adopted 40ms TTT and 10ms TTI for our initial model, so that there will be 4 sub- states in S and M. We have discussed that longer TTT will further benefit in mitigating ping- pong handover. In our model, this will be represented from two aspects: 1) number of states. Since TTI is defined to be 10ms, longer TTT means more sub-states. With same transition probability, more sub-states means it is harder to jump out of the states chain of S or M. UE will be less likely to behave ping-pong handover. 2) transition probability. Once the MCP structure of TTT is fixed, CRE will affect the transition probability to increase PM(x) and reduce PS(x) to help UE offloaded to Small cell states. As a result, with the combination of TTT and CRE, UE will prefer to be offloaded to small cell and remain stable. The detailed deduction and result will be shown in next part. Since ping-pong Handover represents the frequent serving cell switching between macro cell and small cell, we have modelled the Handover loop as into the Markov Chain with n=4 shown in Figure 5.1.

Figure 5-1 Handover model with Markov Chain

After the relationship of all states are defined, we should define the transition probability so that the Markov transit matrix can be established. Suppose UE is bounded to a macro cell and in M1 state, according to UE’s location and RSS. It has probability PM(x) to reach M2 and 1 − PM(x) return M1 to renew the Handover assessment and the TTT count time is set to 0. M2, M3, … Mn states follow the same protocol. Until Mn reaches I1, which means the decision condition is satisfied for the whole TTT, the process phase will be started. During this phase, Handover is guaranteed to happen and only control signal and acknowledge signals are transferred to finish Handover process. Therefore, it has probability of 1 moving from I1 to I2 and then to I3 till Handover finishes and reaches S1 states (handover failure is not considered in this paper, so the probability of handover execution phase is set to be 1).

UE will then renew handover process with probability P_S(x) till next handover. During handover process states (I and I' states), mainly traffic signals are transferred and information signal will be blocked till handover finishes. Therefore, frequent ping-pong handover will significantly reduce UE capacity. The virtual bias CRE will take effect to reduce ping-pong handover and increase capacity accordingly, which may be represented in 𝑃_𝑀(𝑥) and 𝑃𝑆(𝑥) combined with step number and mobility model. The detailed calculation will be discussed in next session.

After defining MC process and its state transit probability, we will be able to establish the transfer Matrix (T) to model the handover process for UEs and analyse how CRE affects it. Table.1 illustrates the 4-state transfer matrix with our algorithm, which will be used in later simulation:

Table 5-1 Markov Transfer Matrix (T)

M1 M2 M3 M4 I1 I2 I3 I4 S1 S2 S3 S4 I1' I2' I3' I4'

M1 1-PM(x) PM(x) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 M2 1-PM(x) 0 PM(x) 0 0 0 0 0 0 0 0 0 0 0 0 0 M3 1-PM(x) 0 0 PM(x) 0 0 0 0 0 0 0 0 0 0 0 0 M4 1-PM(x) 0 0 0 PM(x) 0 0 0 0 0 0 0 0 0 0 0 I1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 I2 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 I3 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 I4 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 S1 0 0 0 0 0 0 0 0 _1-P S(x) PS(x) 0 0 0 0 0 0

S2 0 0 0 0 0 0 0 0 _1-P S(x) 0 PS(x) 0 0 0 0 0 S3 0 0 0 0 0 0 0 0 _1-P S(x) 0 0 PS(x) 0 0 0 0 S4 0 0 0 0 0 0 0 0 _1-P S(x) 0 0 0 PS(x) 0 0 0 I1' 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 I2' 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 I3' 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 I4' 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

The row of Matrix represents ‘start’ state, which means the system is currently in this state. The column for each row means the next state if the ‘start’ state is the row name. Therefore, the value of matrix represents the transition probability from row name to column name. For row M1 to M4, it has probability of 1- PM(x) to reach the column M1, which means jumping

back to the initial state. Meanwhile, it has the probability of PM(x) moving to next state. As

a result, the sum of each row should equals to 1, which means all the possible next states have been taken into consideration. The situation is similar from row S1 to S4. I and I’ row represents the process phase, during which UE is inevitably transferred to the other cell. Therefore, the probability of moving to next state is set to be 1. Still, the total probability of each row is 1 during these two kind of states.

Once transfer matrix is established, the state probability vector 𝑉 at any given step 𝑥 can be calculated according to the property of MC:

𝑉(𝑥) = 𝑉(1) ∏ 𝑇(𝑖) 𝑥

𝑖=1

𝑉(1) shows the probability of UE at initial point, which is given and can be decided with probability of 1. For example, UE is bound to macro cell from time 1, its probability vector will be [1, 0, 0 ... 0]. With time passing by, number of step will increase, its vector will keep changing according to Equation (6). Therefore, we can find out the vectors for all UEs in the system after x steps and analyse their handover rates through the probability of each state.