Metastability in loss networks with dynamic alternative routing
S Olesker-Taylor - The Annals of Applied Probability, 2022 - projecteuclid.org
The Annals of Applied Probability, 2022•projecteuclid.org
Consider N stations interconnected with links, each of capacity K, forming a complete graph.
Calls arrive to each link at rate λ and depart at rate 1. If a call arrives to a link xy, connecting
stations x and y, which is at capacity, then a third station z is chosen uniformly at random and
the call is attempted to be routed via z: if both links xz and zy have spare capacity, then the
call is held simultaneously on these two; otherwise the call is lost. We analyse an
approximation of this model. We show rigorously that there are three phases according to …
Calls arrive to each link at rate λ and depart at rate 1. If a call arrives to a link xy, connecting
stations x and y, which is at capacity, then a third station z is chosen uniformly at random and
the call is attempted to be routed via z: if both links xz and zy have spare capacity, then the
call is held simultaneously on these two; otherwise the call is lost. We analyse an
approximation of this model. We show rigorously that there are three phases according to …
Consider N stations interconnected with links, each of capacity K, forming a complete graph. Calls arrive to each link at rate λ and depart at rate 1. If a call arrives to a link , connecting stations x and y, which is at capacity, then a third station z is chosen uniformly at random and the call is attempted to be routed via z: if both links and have spare capacity, then the call is held simultaneously on these two; otherwise the call is lost.
We analyse an approximation of this model. We show rigorously that there are three phases according to the traffic intensity : for α∈(0,αc)∪(1,∞), the system has mixing time logarithmic in the number of links n:=N2; for α∈(αc,1) the system has mixing time exponential in n, the number of links. Here αc:=13(5 10−13)≈0.937 is an explicit critical threshold with a simple interpretation. We also consider allowing multiple rerouting attempts. This has little effect on the overall behaviour; it does not remove the metastability phase.
Finally, we add trunk reservation: in this, some number σ of circuits are reserved; a rerouting attempt is only accepted if at least circuits are available. We show that if σ is chosen sufficiently large, depending only on α, not K or n, then the metastability phase is removed.
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