Queues with. Session. Arrivals. Introduction. Model. Analysis. Packet delay. Examples. Conclusions. System Content and Packet Delay in Discrete-Time Queues.
System Content and Packet Delay in Discrete-Time Queues with Session-Based Arrivals
Queues with Session Arrivals
Introduction Model Analysis Packet delay Examples
L. Hoflack, S. De Vuyst, S. Wittevrongel, H. Bruneel
Research Group Department of Telecommunications and Information Processing Ghent University, Belgium
Conclusions
‘Session-based’ arrivals? Queues with Session Arrivals
Mathematical model and system equations Introduction Model
Steady-state buffer analysis
Analysis Packet delay Examples
Packet delay Numerical examples Conclusions
Conclusions
Session-based arrivals Queues with Session Arrivals
Traffic generated by ‘users’
Introduction Terminology Application Model Analysis Packet delay Examples Conclusions
Session-based arrivals Queues with Session Arrivals
Traffic generated by ‘users’
Introduction Terminology Application Model Analysis Packet delay
time (slots)
Examples Conclusions
Model in discrete time −→ time is divided in fixed-length intervals: slots
Session-based arrivals Queues with Session Arrivals
Traffic generated by ‘users’
Introduction Terminology Application Model Analysis Packet delay
time (slots)
Examples Conclusions
Model as independent batch arrivals
Session-based arrivals Queues with Session Arrivals
Traffic generated by ‘users’
Introduction Terminology Application Model Analysis Packet delay
time (slots)
Examples Conclusions
Model as train arrivals
Session-based arrivals Queues with Session Arrivals
Traffic generated by ‘users’
Introduction Terminology Application Model Analysis Packet delay
time (slots)
Examples Conclusions
Model as session arrivals
Session-based arrivals Queues with Session Arrivals
Traffic generated by ‘users’
Introduction Terminology Application
Session bandwidth
Model
Session length
Analysis Packet delay
time (slots)
Examples Conclusions
Model as session arrivals
Application Queues with Session Arrivals
Introduction Terminology Application
packets ... File server output buffer
Model
... output channel
Internet
Analysis Packet delay Examples Conclusions
Traffic to output buffer of a file server is session-based one file download y one session
Application Queues with Session Arrivals
Introduction Terminology Application
packets ... File server output buffer
Model
... output channel
Internet
Analysis Packet delay Examples Conclusions
Traffic to output buffer of a file server is session-based one file download y one session −→ Performance analysis of output buffer
Modeling assumptions Queues with Session Arrivals
I
Discrete-time queueing system −→ time is slotted
I
Infinite storage capacity
I
One output line
I
FCFS for packets
I
Transmission time of the packets: geometric distribution
Introduction Model Assumptions Arrival process System equations Analysis
−→ has probability generating function (pgf)
Packet delay Examples Conclusions
σz T (z) = 1 − (1−σ)z In each slot, the packet in service leaves with probability σ
Arrival process: session-based packet arrivals
I
Number of new sessions per slot: pgf S(z)
I
Session length has geometric distribution Prob[session length = n slots] = (1−α)αn−1 ,
Queues with Session Arrivals
Introduction
n>1
(1−α)z pgf L(z) = 1 − αz In each slot, an active session ends with probability 1−α I
Session bandwidth: pgf P (z) requirement: P (0) = 0 number of packets generated in a slot during the session
I
The packet arrivals (from all active sessions) in a slot are buffered in random order
Model Assumptions Arrival process System equations Analysis Packet delay Examples Conclusions
System equations Consider arbitrary slot k: I
Queues with Session Arrivals
sk new sessions starting in slot k pgf S(z)
I
Introduction Model
ak active sessions in slot k ak−1
ak = sk +
X
( cik ,
cik =
i=1 I
1 0
w.p. α w.p. 1−α
Analysis Packet delay
uk buffer content after slot k uk = max(uk−1 − rk , 0) +
Assumptions Arrival process System equations
Examples
ak X
pik ,
pik has pgf P (z)
i=1
( rk =
1 0
‘System’ state is hak , uk i −→ forms a Markov chain !
w.p. σ w.p. 1−σ
Conclusions
Steady-state buffer analysis Joint pgf of the system state:
Qk (x, z) , E[xak z uk ] Queues with Session Arrivals
Relate Qk (x, z) to Qk−1 (x, z) using system equations In steady state (k → ∞), both converge to Q(x, z)
Introduction Model
Functional equation (FE) for function Q(x, z): Q(x, z) =
S(xP (z)) h
(σ + (1−σ)z)Q(1−α + αxP (z), z) + σ(z−1)p0 z p0 : probability of empty buffer determine from normalization condition
Analysis
i
Functional equation Moments buffer content Pgf Tail Packet delay Examples Conclusions
Marginal distributions I I
A(x) = Q(x, 1) U (z) = Q(1, z)
number of active sessions during a slot buffer content after a slot
Moments of the buffer content Does not require to solve the FE directly! i S(xP (z)) h Q(x, z) = (σ + (1−σ)z) Q(1−α + αxP (z), z) +σ(z−1)p0 | {z } | {z } z
Queues with Session Arrivals
Solve only in 1-dimensional subspace where arguments of Q(·, ·) on both
Introduction
sides of the FE are equal
Model
x = 1−α + αxP (z)
⇒
1−α x= 1 − αP (z)
FE solved for these arguments: Q
1−α σ(z − 1)p0 S(L(P (z))) ,z = 1 − αP (z) z − (σ + (1 − σ)z)S(L(P (z)))
Mean buffer content First derivative for z = 1 yields mean buffer content U 0 (1) Subsequent derivatives in z = 1 yield higher-order moments (e.g. variance)
Analysis Functional equation Moments buffer content Pgf Tail Packet delay Examples Conclusions
Pgf of the buffer content Obtain U (z) by successive application of the FE I
Define Ni (z) , 1−α + αP (z)Ni−1 (z) Ni (z) = . . . =
I I
1−α+α(1−P (z))(αP (z))i 1−αP (z)
In FE, put x = 1 Iteratively, U (z) =
∞ Y σ(z − 1)p0 S(L(P (z))) S(P (z)Ni (z))(σ + (1−σ)z) z − (σ + (1−σ)z)S(L(P (z))) i=0 z ∞ i σ(z − 1)p0 X Y S(P (z)Nj (z))(σ + (1−σ)z) + σ + (1−σ)z i=0 j=0 z
Infinite sums and products converge
Queues with Session Arrivals
Introduction Model Analysis Functional equation Moments buffer content Pgf Tail Packet delay Examples Conclusions
Tail distribution of the buffer content What is buffer overflow probability Prob[u > T ]? −→ use dominant pole approximation on the function U (z) Prob[u > T ] ≈ −
B 1 T z0 −1 z0 +1
Introduction
(T large)
where I z0 : smallest real root larger than 1 of denominator U (z) I
B is residue of U (z) in z0 :
Queues with Session Arrivals
B = lim (z − z0 )U (z) z→z0
Model Analysis Functional equation Moments buffer content Pgf Tail Packet delay Examples Conclusions
U (z) =
σ(z − 1)p0 S(L(P (z))) z − (σ + (1−σ)z)S(L(P (z)))
∞ Y i=0
S(P (z)Ni (z))(σ + (1−σ)z) z
∞ i σ(z − 1)p0 X Y S(P (z)Nj (z))(σ + (1−σ)z) + σ + (1−σ)z i=0 j=0 z
Distribution of the packet delay d Consider arbitrary packet P Queues with Session Arrivals
Introduction Model Analysis Packet delay Examples Conclusions
Distribution of the packet delay d Consider arbitrary packet P Queues with Session Arrivals
Introduction
packet P
Model Analysis Packet delay Examples Conclusions
Distribution of the packet delay d Consider arbitrary packet P Queues with Session Arrivals
Introduction Model Analysis Packet delay Examples Conclusions
time (slots)
packet P
Distribution of the packet delay d Consider arbitrary packet P Queues with Session Arrivals
Introduction Model Analysis Packet delay Examples Conclusions
...
...
transmission of P
Distribution of the packet delay d Consider arbitrary packet P Queues with Session Arrivals
Introduction Model Analysis Packet delay
delay d of packet P
Examples Conclusions
...
...
transmission of P
Distribution of the packet delay d Queues with Session Arrivals
Pgf D(z) of packet delay Introduction Model Analysis Packet delay
−→ from general relationship for G/Geom/1 queues D(z) =
σ U (T (z)) − p0 P 0 (1)A0 (1)
Examples Conclusions
Derive mean packet delay D0 (1), higher-order moments and tail distribution
Example: influence of the session bandwidth P (z) Sessions I
A new session starts each slot with probability β: S(z) = 1 − β + βz
I
Queues with Session Arrivals
(Bernoulli)
Introduction
Number of packets generated per slot (session bandwidth):
Model Analysis
P (z) = z I
p
Mean session length:
0
P (1) = p
(deterministic) 1 Ts = L0 (1) = 1−α
Packet delay Examples Session bandwidth Sessions starts Conclusions
Packet transmission time mean value 1/σ
(geometric)
System load −→ average amount of arriving ‘work’ per slot : ρ = β p Ts we change load by varying mean transmission time 1/σ
1 σ
Example: influence of P (z) on buffer content 100 90
Queues with Session Arrivals
T = 2 slots, β = 0.005 s
Mean buffer occupancy U’(1)
80 Introduction
p=1
70
Model
p=2 60
p=4
50
p=8
Analysis Packet delay Examples Session bandwidth Sessions starts
40 30
Conclusions
20 10 0 0
0.2
0.4 0.6 Mean load
0.8
1
Mean buffer content U 0 (1) versus load ρ for session bandwidth p = 1, 2, 4, 8 Gray curves → uncorrelated arrivals (!)
Example: influence of P (z) on packet delay 400 350
Queues with Session Arrivals
T = 2 slots, β = 0.005 s
Mean packet delay
300
p=1
Introduction Model
p=2
250
Analysis
p=4
Packet delay
200
Examples
p=8
Session bandwidth Sessions starts
150
Conclusions
100 50 0 0
0.2
0.4 Mean load
0.6
0.8
Mean packet delay D0 (1) versus load ρ for session bandwidth p = 1, 2, 4, 8
Example: influence of new session distribution S(z) Sessions I
S(z) = 1−γ + γz q
New sessions started per slot:
In each slot, q new sessions start with probability γ
q % then variance % I
Number of packets generated per slot (session bandwidth): P (z) = z p
I
Mean session length:
P 0 (1) = p
(deterministic) 1 Ts = L0 (1) = 1−α
Queues with Session Arrivals
Introduction Model Analysis Packet delay Examples Session bandwidth Sessions starts Conclusions
Packet transmission time mean value 1/σ
System load
(geometric)
ρ = S 0 (1) p Ts
1 σ
we change load by varying mean transmission time 1/σ S 0 (1) = qγ is kept constant at 0.1
Example: influence of S(z) on the buffer content 100 90
Queues with Session Arrivals
T = 2 slots, P’(1)=3, S’(1)= 0.1, σ2 = 0 s
P
Mean buffer occupancy U’(1)
80 Introduction
70
Model Analysis
60
Packet delay Examples
50 q=8
40
Session bandwidth Sessions starts
q=1 q=2
30
Conclusions
q=3 20
q=4
10 0
0.65
0.7
0.75 0.8 0.85 Mean load
0.9
0.95
1
Mean buffer occupancy U 0 (1) versus load ρ for q = 1, 2, 3, 4, 8.
Conclusions Queues with Session Arrivals
I
Analysis of single-server discrete-time queue with session-based arrivals and geometric service times
Introduction Model Analysis Packet delay
I
Derived expressions for pgf of buffer content and pgf of packet delay −→ moments and tail distribution
I
Buffer performance can be severely underestimated if correlation due to sessions is not taken into account
Examples Conclusions
