Metropolis Markov chains for wireless random-access networks - - PowerPoint PPT Presentation

Metropolis Markov chains for wireless random-access networks

Alessandro Zocca

joint work with Sem C. Borst, Johan S. H. van Leeuwaarden, Francesca R. Nardi

Berlin, October 24th, 2014

1

Conflict graph

N sites

2

Conflict graph

Hard-core constraints → edges N

2

Conflict graph

Hard-core constraints → edges N

2

Conflict graph

Hard-core constraints → edges N

2

Conflict graph

Undirected graph G = (V , E) with N vertices

2

Conflict graph

Presence of a particle → occupied site N

2

Admissible configurations

Any admissible configuration can be represented by a vector x = (x1, . . . , xN) ∈ {0, 1}N, where xi = 1 ⇐ ⇒ site i is occupied.

3

Admissible configurations

Any admissible configuration can be represented by a vector x = (x1, . . . , xN) ∈ {0, 1}N, where xi = 1 ⇐ ⇒ site i is occupied. The set of admissible configurations is Ω(G) = {x ∈ {0, 1}N : xi + xj ≤ 1 ∀ (i, j) ∈ E}

3

Admissible configurations

Any admissible configuration can be represented by a vector x = (x1, . . . , xN) ∈ {0, 1}N, where xi = 1 ⇐ ⇒ site i is occupied. The set of admissible configurations is Ω(G) = {x ∈ {0, 1}N : xi + xj ≤ 1 ∀ (i, j) ∈ E} Ω(G) ≡ I(G) = {I ⊂ V : I independent set}

3

Conflict graph for wireless networks

N devices/users

4

Conflict graph for wireless networks

Edges → interference constraints N

4

Conflict graph for wireless networks

Edges → interference constraints N

4

Conflict graph for wireless networks

G = (V , E) → wireless network contention graph

4

Conflict graph for wireless networks

Occupied site → transmitting nodes N

4

Random-access wireless networks

Randomized medium-access algorithms provide popular mechanism for distributed medium-access control with low implementation complexity

5

Random-access wireless networks

Randomized medium-access algorithms provide popular mechanism for distributed medium-access control with low implementation complexity Carrier-Sense Multiple-Access (CSMA)

5

Random-access wireless networks

Randomized medium-access algorithms provide popular mechanism for distributed medium-access control with low implementation complexity Carrier-Sense Multiple-Access (CSMA) Each node obeys random back-off time before starting transmission

• back-off timer frozen while medium is sensed busy
• new back-off timer is set after each transmission

5

Random-access wireless networks

Randomized medium-access algorithms provide popular mechanism for distributed medium-access control with low implementation complexity Carrier-Sense Multiple-Access (CSMA) Each node obeys random back-off time before starting transmission

• back-off timer frozen while medium is sensed busy
• new back-off timer is set after each transmission

Main goal

Study the effect of the network topology on its delay performance

5

Local dynamics

Back-off period is “frozen” if the node is blocked Exp(1)

6

Local dynamics

Back-off period is “frozen” if the node is blocked Exp(1)

6

Local dynamics

Back-off periods at each node are i.i.d. Exp(ν)

6

Local dynamics

Back-off periods at each node are i.i.d. Exp(ν)

6

Local dynamics

Packet transmission times at each node are i.i.d. Exp(1)

6

Local dynamics

Packet transmission times at each node are i.i.d. Exp(1)

6

Markov process

Let Xt ∈ Ω represent the network activity configuration at time t.

7

Markov process

Let Xt ∈ Ω represent the network activity configuration at time t. (Xt)t≥0 is a reversible continuous-time Markov process on Ω with transition rates q(x, y) =    ν if y = x + ei ∈ Ω, 1 if y = x − ei ∈ Ω,

• therwise.

7

Markov process

Let Xt ∈ Ω represent the network activity configuration at time t. (Xt)t≥0 is a reversible continuous-time Markov process on Ω with transition rates q(x, y) =    ν if y = x + ei ∈ Ω, 1 if y = x − ei ∈ Ω,

• therwise.

The Markov process (Xt)t≥0 is irreducible and positive recurrent on Ω

7

Markov process

Let Xt ∈ Ω represent the network activity configuration at time t. (Xt)t≥0 is a reversible continuous-time Markov process on Ω with transition rates q(x, y) =    ν if y = x + ei ∈ Ω, 1 if y = x − ei ∈ Ω,

• therwise.

The Markov process (Xt)t≥0 is irreducible and positive recurrent on Ω Product-form stationary distribution π(x) = Z −1ν

N

i=1 xi,

x ∈ Ω

7

Long-term vs. short-term fairness

Product-form stationary distribution of (Xt)t≥0 π(x) = Z −1ν

N

i=1 xi,

x ∈ Ω

8

Long-term vs. short-term fairness

Product-form stationary distribution of (Xt)t≥0 π(x) = Z −1ν

N

i=1 xi,

x ∈ Ω As ν grows large, the configurations with most active nodes attract most

• f the probability mass

8

Long-term vs. short-term fairness

Product-form stationary distribution of (Xt)t≥0 π(x) = Z −1ν

N

i=1 xi,

x ∈ Ω As ν grows large, the configurations with most active nodes attract most

• f the probability mass

However these maximum-size transmission patterns are “rigid”

8

Long-term vs. short-term fairness

Product-form stationary distribution of (Xt)t≥0 π(x) = Z −1ν

N

i=1 xi,

x ∈ Ω As ν grows large, the configurations with most active nodes attract most

• f the probability mass

However these maximum-size transmission patterns are “rigid”

Regime of interest: ν → ∞

8

Transition times and mixing times

Dominant configurations = maximum/maximal independent sets of G

9

Transition times and mixing times

Dominant configurations = maximum/maximal independent sets of G We study the asymptotic behavior as ν → ∞ of

9

Transition times and mixing times

Dominant configurations = maximum/maximal independent sets of G We study the asymptotic behavior as ν → ∞ of

• Transition time T(ν) between dominant configurations as ν → ∞

9

Transition times and mixing times

Dominant configurations = maximum/maximal independent sets of G We study the asymptotic behavior as ν → ∞ of

• Transition time T(ν) between dominant configurations as ν → ∞
• Mixing time tmix(ε, ν) of the activity process (Xt)t≥0 as ν → ∞

9

Complete K-partite networks [ZBvL12]

G = K4,3,5,2,6

10

Complete K-partite networks [ZBvL12]

K = 5, (L1, L2, L3, L4, L5) = (4, 3, 5, 2, 6)

10

Complete K-partite networks [ZBvL12]

State space Ω

10

Complete K-partite networks [ZBvL12]

State space aggregation: Ω (left) and Ω∗ (right)

10

Complete K-partite networks [ZBvL12]

Aggregated state space Ω∗

Ω∗ = {0} ∪ {(k, l) ∈ N2 : 1 ≤ k ≤ K, 1 ≤ l ≤ Lk},

10

State space aggregation

The state aggregation yields a new Markov process (X ∗

t )t≥0 on

Ω∗ = {0} ∪ {(k, l) ∈ N2 : 1 ≤ k ≤ K, 1 ≤ l ≤ Lk},

11

State space aggregation

The state aggregation yields a new Markov process (X ∗

t )t≥0 on

Ω∗ = {0} ∪ {(k, l) ∈ N2 : 1 ≤ k ≤ K, 1 ≤ l ≤ Lk}, with transition rates q ((k, l), (k, l + 1)) = (Lk − l)ν, for 1 ≤ l < Lk q (0, (k, 1)) = Lkν, q ((k, l), (k, l − 1)) = l, for 1 < l ≤ Lk q ((k, 1), 0) = 1,

11

State space aggregation

The state aggregation yields a new Markov process (X ∗

t )t≥0 on

Ω∗ = {0} ∪ {(k, l) ∈ N2 : 1 ≤ k ≤ K, 1 ≤ l ≤ Lk}, with transition rates q ((k, l), (k, l + 1)) = (Lk − l)ν, for 1 ≤ l < Lk q (0, (k, 1)) = Lkν, q ((k, l), (k, l − 1)) = l, for 1 < l ≤ Lk q ((k, 1), 0) = 1, and stationary distribution π(k,l)(ν) = π0(ν) Lk l

• νl,

l = 1, . . . , Lk, k = 1, . . . , K, π0(ν) =

• 1 +

K

• k=1

Lk

• l=1

Lk l

• νl

−1

11

Dominant configurations

The maximal independent sets of G corresponds to the configurations si = (ki, Li) ∈ Ω∗, i = 1, . . . , K

12

Dominant configurations

The maximal independent sets of G corresponds to the configurations si = (ki, Li) ∈ Ω∗, i = 1, . . . , K si is also maximum independent set if Li = maxk Lk

12

Asymptotic escape time T(k,l)→0 from branch k

Proposition (Asymptotics for the escape time)

For any 0 < l ≤ Lk, ET(k,l)→0(ν) ∼ 1 Lk νLk−1, as ν → ∞, and T(k,l)→0(ν) ET(k,l)→0(ν)

d

− → Exp(1), as ν → ∞.

13

Asymptotic escape time T(k,l)→0 from branch k

Proposition (Asymptotics for the escape time)

For any 0 < l ≤ Lk, ET(k,l)→0(ν) ∼ 1 Lk νLk−1, as ν → ∞, and T(k,l)→0(ν) ET(k,l)→0(ν)

d

− → Exp(1), as ν → ∞. Remark: no dependence on the initial number l of active nodes!

13

Stochastic representation for the transition time Ts1→s2

Ts1→s2

d

= Ts1→0 +

• k=2

Nk

• i=1
• ˆ

T (i)

0→(k,1) + T (i) (k,1)→0

• + ˆ

T0→s2. where Nk

d

= Geo

• L2

L2 + Lk

• 14

Stochastic representation for the transition time Ts1→s2

Ts1→s2

d

= Ts1→0 +

• k=2

Nk

• i=1
• ˆ

T (i)

0→(k,1) + T (i) (k,1)→0

• + ˆ

T0→s2. where Nk

d

= Geo

• L2

L2 + Lk

• 14

Stochastic representation for the transition time Ts1→s2

Ts1→s2 ≈ Ts1→0 +

• k=2

Nk

• i=1
• ˆ

T (i)

0→(k,1) + T (i) (k,1)→0

• + ˆ

T0→s2. where Nk

d

= Geo

• L2

L2 + Lk

• 14

Stochastic representation for the transition time Ts1→s2

Ts1→s2 ≈ Ts1→0 +

• k=2

Nk

• i=1
• ˆ

T (i)

0→(k,1) + T (i) (k,1)→0

• + ˆ

T0→s2. where Nk

d

= Geo

• L2

L2 + Lk

• 14

Stochastic representation for the transition time Ts1→s2

Ts1→s2 ≈ Ts1→0 +

• k=2

Nk

• i=1
• ˆ

T (i)

0→(k,1) + T (i) (k,1)→0

• + ˆ

T0→s2. where Nk

d

= Geo

• L2

L2 + Lk

• Define L∗ := max

k=2 Lk and K∗ = {k = 2 : Lk = L∗}

14

Stochastic representation for the transition time Ts1→s2

Ts1→s2 ≈ Ts1→0 +

• k∈K∗

Nk

• i=1
• ˆ

T (i)

0→(k,1) + T (i) (k,1)→0

• + ˆ

T0→s2. where Nk

d

= Geo

• L2

L2 + Lk

• Define L∗ := max

k=2 Lk and K∗ = {k = 2 : Lk = L∗}

14

Stochastic representation for the transition time Ts1→s2

Ts1→s2 ≈ Ts1→0 +

M

• i=1
• ˆ

T (i)

0→(k,1) + T (i) (k,1)→0

• + ˆ

T0→s2. where Nk

d

= Geo

• L2

L2 + Lk

• .

Define L∗ := max

k=2 Lk and K∗ = {k = 2 : Lk = L∗}

M d = Geo

• |K∗|L∗

|K∗|L∗ + L2

• .

14

Asymptotics for the transition time Ts1→s2

Theorem

ETs1→s2(ν) ∼ 1{s1∈K∗} L∗ + |K∗| L2

• νL∗−1,

as ν → ∞, and

15

Asymptotics for the transition time Ts1→s2

Theorem

ETs1→s2(ν) ∼ 1{s1∈K∗} L∗ + |K∗| L2

• νL∗−1,

as ν → ∞, and Ts1→s2(ν) ETs1→s2(ν)

d

− → 1 EM

M

• i=1

Yi, as ν → ∞, where M d = Geo(p∗) + 1{s1∈K∗}, p∗ =

|K∗|L∗ |K∗|L∗+L2 , and Yi are i.i.d. Exp(1).

15

Asymptotics for the transition time Ts1→s2

Theorem

ETs1→s2(ν) ∼ 1{s1∈K∗} L∗ + |K∗| L2

• νL∗−1,

as ν → ∞, and Ts1→s2(ν) ETs1→s2(ν)

d

− → 1 EM

M

• i=1

Yi, as ν → ∞, where M d = Geo(p∗) + 1{s1∈K∗}, p∗ =

|K∗|L∗ |K∗|L∗+L2 , and Yi are i.i.d. Exp(1).

In particular, s1 ∈ K∗ = ⇒ Ts1→s2(ν) ETs1→s2(ν)

d

− → Exp(1), as ν → ∞.

15

Mixing time

16

Mixing time

Relabel the components such that L1 ≥ L2 ≥ · · · ≥ LK.

16

Mixing time

Relabel the components such that L1 ≥ L2 ≥ · · · ≥ LK.

Theorem

tmix(ε, ν) = Θ(νL2−1), ν → ∞.

16

Mixing time

Relabel the components such that L1 ≥ L2 ≥ · · · ≥ LK.

Theorem

tmix(ε, ν) = Θ(νL2−1), ν → ∞. Idea of the proof:

16

Mixing time

Relabel the components such that L1 ≥ L2 ≥ · · · ≥ LK.

Theorem

tmix(ε, ν) = Θ(νL2−1), ν → ∞. Idea of the proof:

• Upper bound by coupling argument

16

Mixing time

Relabel the components such that L1 ≥ L2 ≥ · · · ≥ LK.

Theorem

tmix(ε, ν) = Θ(νL2−1), ν → ∞. Idea of the proof:

• Upper bound by coupling argument
• Lower bound by giving an upper bound for the conductance

(bottleneck ratio) Φ = min

S⊂Ω:π(S)≤1/2 Q(S, Sc)/π(S)

16

Heterogeneous complete K-partite graph [ZBvL14]

Heterogeneity: users in component k have a back-off rate fk(ν).

17

Heterogeneous complete K-partite graph [ZBvL14]

Heterogeneity: users in component k have a back-off rate fk(ν).

Asymptotics for the transition time Ts1→s2 (part I)

ETs1→s2(ν) ∼ 1 Ls1 fs1(ν)Ls1−1 + 1 Ls2fs2(ν)

• k∈K∗

fk(ν)Lk, as ν → ∞.

17

Heterogeneous complete K-partite graph [ZBvL14]

Heterogeneity: users in component k have a back-off rate fk(ν).

Asymptotics for the transition time Ts1→s2 (part I)

ETs1→s2(ν) ∼ 1 Ls1 fs1(ν)Ls1−1 + 1 Ls2fs2(ν)

• k∈K∗

fk(ν)Lk, as ν → ∞. where γk := limν→∞

fk(ν)Lk

• j=s2 fj(ν)Lj and K∗ := {k = 2 : γk > 0}.

17

Heterogeneous complete K-partite graph [ZBvL14]

Heterogeneity: users in component k have a back-off rate fk(ν).

Asymptotics for the transition time Ts1→s2 (part II)

Ts1→s2(ν) ETs1→s2(ν)

d

− → Z = αY + (1 − α)W , as ν → ∞, where Y d = Exp(1) and LW (s) =

• 1 +
• k∈A

γks 1 + γks/βk + s

• k∈S

γk −1 .

18

Heterogeneous complete K-partite graph [ZBvL14]

Heterogeneity: users in component k have a back-off rate fk(ν).

Asymptotics for the transition time Ts1→s2 (part II)

Ts1→s2(ν) ETs1→s2(ν)

d

− → Z = αY + (1 − α)W , as ν → ∞, where Y d = Exp(1) and LW (s) =

• 1 +
• k∈A

γks 1 + γks/βk + s

• k∈S

γk −1 . where βk := limν→∞

Lkfk(ν) Ls2fs2(ν) and K∗ is partitioned into three subsets

• k ∈ N if βk = 0,
• k ∈ A if βk ∈ R+,
• k ∈ S if βk = ∞.

18

Heterogeneous complete K-partite graph [ZBvL14]

Possible asymptotic distribution for Z

α A S Limiting distribution Z ∅ ∅ δ0 (trivial r.v. identical to 0) n.e. ∅ G

i=1 Hi( β1 βA , . . . , βm βA , β1 γ1 , . . . , βm γm )

G

i=1 Expi(λ)

if βk/γk = λ ∀ k ∈ A ∅ n.e. Exp(1/γS) n.e. n.e. W (0, 1) ∅ ∅ Exp(1/α) Exp(1/α) + G

i=1 Hi

• β1

βA , . . . , βm βA , β1 (1−α)γ1 , . . . , βm (1−α)γm

• n.e.

∅ Exp(1/α) + G

i=1 Expi

• λ/(1 − α)
• if βk/γk = λ

∀ k ∈ A Exp

• α−1(1 + m

k=1 βk)−1

if βk/γk = (1 − α)/α ∀ k ∈ A Exp(1) if βk/γk = (1 − α)/α = m

i=1 βk

∀ k ∈ A ∅ n.e. Exp(1/α) + Exp(1/(1 − α)γS) Erlang(2, 1/α) if α = γS/(1 + γS) n.e. n.e. Exp(1/α) + (1 − α)W 1

• Exp(1)

19

Grid graphs [ZBvLN13]

20

Grid graphs [ZBvLN13]

“even chessboard” E

20

Grid graphs [ZBvLN13]

“odd chessboard” O

20

Boundary conditions

2K × 2L open grid networks

21

Boundary conditions

2K × 2L toric grid networks

21

Boundary conditions

2K × 2L cylindrical grid networks

21

Transition and mixing times results

Let G be a grid graph

22

Transition and mixing times results

Let G be a grid graph

Theorem (Mean transition time)

ETE→O(ν) νΓ(G)−1, as ν → ∞.

22

Transition and mixing times results

Let G be a grid graph

Theorem (Mean transition time)

ETE→O(ν) νΓ(G)−1, as ν → ∞. lim inf

ν→∞

log ETE→O(ν) log ν ≥ Γ(G) − 1

22

Transition and mixing times results

Let G be a grid graph

Theorem (Mean transition time)

ETE→O(ν) νΓ(G)−1, as ν → ∞. lim inf

ν→∞

log ETE→O(ν) log ν ≥ Γ(G) − 1

Theorem (Mixing time)

tmix(ε, ν) ≥ CενΓ(G)−1, as ν → ∞.

22

Analysis of the state space Ω(G)

Key ingredient = analysis of the “bottlenecks” of the state space Ω

23

Analysis of the state space Ω(G)

Key ingredient = analysis of the “bottlenecks” of the state space Ω Intuition During the transition from E to O the process has to visit “mixed configurations” with fewer active nodes

23

Analysis of the state space Ω(G)

Key ingredient = analysis of the “bottlenecks” of the state space Ω Intuition During the transition from E to O the process has to visit “mixed configurations” with fewer active nodes

23

Analysis of the state space Ω(G)

Key ingredient = analysis of the “bottlenecks” of the state space Ω Intuition During the transition from E to O the process has to visit “mixed configurations” with fewer active nodes Bottleneck = subset of low-probability configurations in the middle to go from E to O

23

Where is the bottleneck?

24

Where is the bottleneck?

State space Ω for the complete bipartite grid

24

Where is the bottleneck?

25

Where is the bottleneck?

State space Ω for the 4 × 4 toric grid

25

Bottleneck analysis

In general, the bottleneck is a large and intricate subset of Ω(G), which consists of “mixed configurations”

26

Bottleneck analysis

In general, the bottleneck is a large and intricate subset of Ω(G), which consists of “mixed configurations” ∆(x) := N 2 −

N

• i=1

xi

26

Bottleneck analysis

In general, the bottleneck is a large and intricate subset of Ω(G), which consists of “mixed configurations” ∆(x) := N 2 −

N

• i=1

xi “inefficiency” of x ∈ Ω(G)

26

Bottleneck analysis

In general, the bottleneck is a large and intricate subset of Ω(G), which consists of “mixed configurations” ∆(x) := N 2 −

N

• i=1

xi “inefficiency” of x ∈ Ω(G) To understand how the transition E → O occurs, the key quantity is Γ(G) := min

ω:E→O max x∈ω ∆(x)

efficiency gap

26

Metropolis Markov chains

Setting ν = eβ, the stationary distribution can be rewritten πβ(x) = e−βH(x)

• y∈Ω e−βH(y) ,

x ∈ Ω, where H : Ω → R is defined as H(x) := − N

i=1 xi for every x ∈ Ω.

27

Metropolis Markov chains

Setting ν = eβ, the stationary distribution can be rewritten πβ(x) = e−βH(x)

• y∈Ω e−βH(y) ,

x ∈ Ω, where H : Ω → R is defined as H(x) := − N

i=1 xi for every x ∈ Ω.

After uniformization, the resulting Markov chain has Metropolis transition probabilities Pβ(x, y) =

• q(x, y)e−β[H(y)−H(x)]+,

if x = y, 1 −

z=y Pβ(x, z),

if x = y.

27

Efficiency gap for grid networks

We can rewrite ∆(x) = H(x) − min

y∈Ω H(y) ≥ 0

28

Efficiency gap for grid networks

We can rewrite ∆(x) = H(x) − min

y∈Ω H(y) ≥ 0

and Γ(G) = min

ω:E→O max x∈ω ∆(x),

is the minimum energy barrier to overcome to go from E to O.

28

Efficiency gap for grid networks

We can rewrite ∆(x) = H(x) − min

y∈Ω H(y) ≥ 0

and Γ(G) = min

ω:E→O max x∈ω ∆(x),

is the minimum energy barrier to overcome to go from E to O.

Theorem

Γ(G) =      min{2K, 2L} + 1 if G is a 2K × 2L toric grid, min{K, L} + 1 if G is a 2K × 2L open grid, min{2K, L} + 1 if G is a 2K × 2L cylindrical grid.

28

Proof idea 1

29

Proof idea 1

Lemma (Inefficiency on a band)

∆B(x) = 0 ⇐ ⇒ x|B = E|B or x|B = O|B

29

Proof idea 1

Lemma (Inefficiency on a band)

∆B(x) = 0 ⇐ ⇒ x|B = E|B or x|B = O|B

Lemma (Inefficiency on rows)

∆r(x) = 0 ⇐ ⇒ x|r = E|r or x|r = O|r

29

Proof idea 2 (Lower bound for Γ(G))

30

Proof idea 2 (Lower bound for Γ(G))

30

Proof idea 2 (Lower bound for Γ(G))

30

Recent results

31

Recent results

Theorem (Mean transition time)

lim

ν→∞

log ETE→O(ν) log ν = Γ(G) − 1, as ν → ∞

31

Recent results

Theorem (Mean transition time)

lim

ν→∞

log ETE→O(ν) log ν = Γ(G) − 1, as ν → ∞

Theorem (Asymptotic exponentiality)

TE→O(ν) ETE→O(ν)

d

− → Exp(1), as ν → ∞

31

Packet dynamics: non-saturated buffers

• Saturated buffers
• Non-saturated buffers

32

Packet dynamics: non-saturated buffers

• Saturated buffers
• Non-saturated buffers

32

Packet queue dynamics

Packets arrive at various nodes according to Poisson processes with rate λ

33

Packet queue dynamics

Packets arrive at various nodes according to Poisson processes with rate λ Each node is active a fraction of time ↑ 1/2 as ν → ∞

33

Packet queue dynamics

Packets arrive at various nodes according to Poisson processes with rate λ Each node is active a fraction of time ↑ 1/2 as ν → ∞

Normalized load

ρ = 2λ

33

Packet queue dynamics

Packets arrive at various nodes according to Poisson processes with rate λ Each node is active a fraction of time ↑ 1/2 as ν → ∞

Normalized load

ρ = 2λ

Proposition (Stability)

In order for all queues to be stable, it is required that ν > ρ 2(1 − ρ)

33

A vicious circle

• High load ρ requires a high activation rate ν

34

A vicious circle

• High load ρ requires a high activation rate ν
• High activation rate ν implies extremely slow transitions between the

two dominant configurations E and O

34

A vicious circle

• High load ρ requires a high activation rate ν
• High activation rate ν implies extremely slow transitions between the

two dominant configurations E and O

• Slow transitions cause starvation and hence excessively long queues

and delays

34

A vicious circle

• High load ρ requires a high activation rate ν
• High activation rate ν implies extremely slow transitions between the

two dominant configurations E and O

• Slow transitions cause starvation and hence excessively long queues

and delays

Theorem (Average delay scaling)

EW (ρ)

• 1

1 − ρ Γ(G)−1 as ρ ↑ 1

34

Transition time and delay

Theorem (Long-term average delay)

lim inf

ν→∞

EW (ν) ETE→O(ν) ≥ 1 4 − 2ρ

35

Transition time and delay

Theorem (Long-term average delay)

lim inf

ν→∞

EW (ν) ETE→O(ν) ≥ 1 4 − 2ρ Sketch of the proof

35

A simulation picture

6 × 6 toric grid with load ρ = 0.8

36

Future research

• Multiple antennas and multiple frequencies

37

Future research

• Multiple antennas and multiple frequencies
• Effect of small perturbations of the conflict graph

37

Future research

• Multiple antennas and multiple frequencies
• Effect of small perturbations of the conflict graph
• Analysis of other regular conflict graphs (e.g. triangular and

hexagonal lattices)

37

Future research

• Multiple antennas and multiple frequencies
• Effect of small perturbations of the conflict graph
• Analysis of other regular conflict graphs (e.g. triangular and

hexagonal lattices)

Thank you for your attention

37

Future research

• Multiple antennas and multiple frequencies
• Effect of small perturbations of the conflict graph
• Analysis of other regular conflict graphs (e.g. triangular and

hexagonal lattices)

Thank you for your attention

[ZBvL12] A. Zocca, S.C. Borst, J.S.H. van Leeuwaarden, Mixing properties of CSMA networks on partite graphs, Valuetools 2012 [ZBvLN13] A. Zocca, S.C. Borst, J.S.H. van Leeuwaarden, F.R Nardi, Delay performance in random-access grid networks. Performance Evaluation, 70(10), 2013 [ZBvL14] A. Zocca, S.C. Borst, J.S.H. van Leeuwaarden, Slow transitions, slow mixing and starvation in dense random-access networks, arXiv:1403.3325

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