Distributed IRSs Always Benefit Every Mobile Operator

We investigate the impact of multiple distributed intelligent reflecting surfaces (IRSs), which are deployed and optimized by a mobile operator (MO), on the performance of user equipments (UEs) served by other co-existing out-of-band (OOB) MOs that do not control the IRSs. We show that, under round-robin scheduling, in mmWave frequencies, the ergodic sum spectral efficiency (SE) of an OOB MO increases logarithmically in the total number of IRS elements with a pre-log factor that increases with the ratio of the number of OOB paths through the IRS to the number of elements at an IRS. We further show that the maximum achievable SE of the OOB MO scales log-linearly with the total IRS elements, with a pre-log factor of $1$. Then, we specify the minimum number of IRSs as a function of the channel parameters and design a distributed IRS system in which an OOB MO almost surely obtains the maximum SE. Finally, we prove that the outage probability at an OOB UE decreases exponentially as the number of IRSs increases, even though they are randomly configured from the OOB UE's viewpoint. We numerically verify our theory and conclude that distributed IRSs always help every MO, but the MO controlling the IRSs benefits the most.