This appendix describes the discrete event simulation model for the LTE eSRVCC with limited access transfers. Let τL,0and τU,0 be the residual lives of the LTE residence time tL,i
and the UMTS residence time tU,i, respectively. In the simulation, there are two methods to generate the samples of the residual lives τL,0 and τU,0.
Method 1 actually simulates the inter-call arrival times and the LTE/UMTS residence
times. The samples of the residual lives τL,0 and τU,0 are computed from the time difference between the call arrival and the subsequent access transfer.
Method 2 generates the samples of the residual lives τL,0 and τU,0 from the residual life random number generators (to be elaborated later) [17].
This appendix describes the simulation model based on Method 2. The Method 1 sim-ulation is similar to the one developed in the supplementary document of [15], and the details are omitted. In the remainder of this appendix, we first introduce the theorem for the residual life random number generation [17]. Then we describe our simulation model.
Theorem 1: Let t be a Gamma random variable with the mean E[t] and the variance V . Define t∗ to be a Gamma random variable with the mean E[t]+V /E[t] and the variance V + (V /E[t])2. Let random variable u be uniformly distributed over the interval (0,1).
Let τ be the residual life of t. Then the distribution of τ is the same as the distribution of u × t∗.
In the simulation, suppose that the samples of the call holding time tc are obtained from an exponential generator Gc with the mean 1/µ, and PL is the probability that a call will arrive at the LTE domain. Let G(E[t], V ) be the Gamma random number generator with the mean E[t] and the variance V . The tL,iand tU,i samples are obtained from G(E[tL,i], VL) and G(E[tU,i], VU), respectively. The samples of u are obtained from a Uniform generator Gu. The samples of t∗L and t∗U are obtained from G(E[tL,i] + VL/E[tL,i], VL+ (VL/E[tL,i])2) and G(E[tU,i] + VU/E[tU,i], VU + (VU/E[tU,i])2), respectively. According to Theorem 1, the samples of τL,0 and τU,0 are obtained from the residual life generators GL and GU, which multiply the u samples (generated from Gu) by the t∗L and t∗U samples (generated from G(E[tL,i] + VL/E[tL,i], VL + (VL/E[tL,i])2) and G(E[tU,i] + VU/E[tU,i], VU + (VU/E[tU,i])2), respectively.
We conduct the replicated simulation experiments. In every replicated run, we simulate a call arrival and the access transfers in this call. In the simulation model, an event e has two attributes:
• The type attribute indicates the event type. There are three event types. A U-L Access Transfer event represents that the UE performs an access transfer from UMTS to LTE during a call. An L-U Access Transfer event represents that the UE performs an access transfer from LTE to UMTS during a call. A Call Release event represents that a call is released.
• The ts attribute is the timestamp when the event occurs.
Six variables are used in the simulation:
• n∗: number of U-L access transfers in a call (i.e., in a replicated run)
• n∗s: total number of L-U and U-L access transfers investigated in the simulation (i.e., in all replicated runs)
• Nc: total number of replicated runs (i.e., simulated calls) in the simulation
• TL: portion of the call holding times that the UE resides in LTE
• TU: portion of the call holding times that the UE resides in UMTS
• domain: a flag that indicates the domain (LTE or UMTS) where the UE resides
From the above variables, we compute
E[N∗|N = n] = n∗s/Nc and θ = TL/(TL+ TU)
In the simulation, a clock ck is maintained to indicate the simulation progress, which is
removed from the event list when the event is processed. Initially, ck is set to 0 to represent the call arrival time (t2 in Figure 3). When the call is released (t8 in Figure 3), a replicated run is complete. The clock ck is reset to 0, and the event list is re-initiated for the next replicated run. One million simulation runs are executed to obtain stable results. Figure 7 illustrates the simulation flow chart with the following steps:
Step 1. Set n∗s, Nc, TL, and TU to 0.
Step 2. Initialize the event list. Set n∗ and the simulation clock ck to 0.
Step 3. The domain where the UE resides when a call arrives is determined as follows.
Generate a Uniform random number u which is drawn from Gu. If u < PL, it means that the UE resides in LTE when the call arrives, and Step 4 is executed. Otherwise (i.e., the UE resides in UMTS), Step 5 is executed.
Step 4. The UE resides in LTE when the call arrives. Set domain to LTE. The Call Release event e1 and first L-U Access Transfer event e2 are generated and inserted into the event list. For event e1, e1.type is Call Release and e1.ts is set to ck plus tc
obtained from Gc. For event e2, e2.type is L-U Access Transfer and e2.ts is set to ck plus τL,0 obtained from GL. Then Step 6 is executed.
Step 5. The UE resides in UMTS when the call arrives. Set domain to UMTS. The Call Release event e1 and first U-L Access Transfer event e3 are generated and inserted into the event list. For event e1, e1.type is Call Release and e1.ts is set to ck plus tc
obtained from Gc. For event e3, e3.type is U-L Access Transfer and e3.ts is set to ck plus τU,0 obtained from GU.
Step 6. The first event e in the event list is deleted and is processed based on its type in Step 7.
U-L Access
Step 7. If e.type is U-L Access Transfer, then Step 8 is executed. If e.type is L-U Access Transfer, then Step 12 is executed. If e.type is Call Release, the simulation proceeds to Step 13.
Step 8 (U-L Access Transfer). If n∗ < N, the network performs access transfer from UMTS to LTE, and Step 9 is executed. Otherwise, the transfer limit is reached, the UE has to remain in UMTS, and the simulation proceeds to Step 10.
Step 9. The UE executes the U-L Access Transfer procedure at UMTS. Increment both n∗ and n∗s by 1. Set domain to LTE. The next L-U Access Transfer event e2 is generated and inserted into the event list, where e2.type is L-U Access Transfer and e2.ts is set to e.ts plus tL,i obtained from G(E[tL,i], VL).
Step 10. Calculate the time interval e.ts − ck that the UE resides in UMTS. Increase TU
by this amount.
Step 11. Advance the simulation clock ck to e.ts, and proceed to Step 6.
Step 12 (L-U Access Transfer). The UE executes the L-U Access Transfer procedure at LTE. Increment n∗s by 1. Set domain to UMTS. Calculate the time interval e.ts − ck that the UE resides in LTE and increase TL by this amount. The next U-L Access Transfer event e3 is generated and inserted into the event list, where e3.type is U-L Access Transfer and e3.ts is set to e.ts plus tU,i obtained from G(E[tU,i], VU). The simulation proceeds to Step 11.
Step 13 (Call Release). If domain is UMTS, it means that the UE resides in UMTS when the call is released, and Step 14 is executed. Otherwise (i.e., the UE resides in LTE), the simulation proceeds to Step 15.
Step 14. Calculate the time interval e.ts − ck that the UE resides in UMTS. Increase TU
by this amount. Then the simulation goes to Step 16.
Step 15. Calculate the time interval e.ts − ck that the UE resides in LTE. Increase TL by this amount.
Step 16. A call is released and Nc is incremented by 1.
Step 17. If one million of calls have been processed, then Step 18 is executed. Otherwise, the simulation proceeds to Step 2 for the next replicated run. In our experience, one million of Call Release events are enough to produce stable statistics, where the confidence intervals of the 99% confidence levels are within 3% of the mean values in most cases.
Step 18. The performance measures (i.e., E[N∗|N = n] and θ) are computed, and the simulation is terminated.
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