A gentle introduction to fluid and diffusion limits for queues

Presented by:

Varun Gupta

April 12, 2006

Example : Tandem Queues

• Interarrival times at queue A are i.i.d. random variables• Interarrival times at queue C are no more independent –

they are ‘weakly’ dependent• Very difficult to analyze queues with correlated

input/service processes.

Example : Non-stationary Queues

• The arrival process fluctuates over a day – high load during day, low load during night

• Difficult to analyze queues with 2 environment states• Numerical methods exist if the arrival process has

certain Markovian properties• Exact solutions for more complex environment

processes are intractable

Example : Non-stationary Queues

• Observation: The soujorn times in environment states are much larger than the service and interarrival times

• Question: What is the limiting queue length distribution as the mean environment state sojourn times become infinity?

A model for non-stationary queues

• Define E, a reference environment process, as a random process taking values in {1,2,..,m}.

• En are a family of slowly-changing environment process defined by time scaling E as

• Nn {Nn(t): t 0} is the queue length process obtained by

letting the system evolve as a GI/GI/1 queue with mean arrival rate 1/

i and mean service rate 1/

i when En is in

state i.

Fluid Approximation

Fluid Approximation

The Functional SLLN

• Xi : i.i.d. random variables with mean m, finite variance 2

• Let

Question: How does the plot of first n partial sums behave as n increases?

n=10

n=100

n=10000

The Functional SLLN

• Define the continuous parameter stochastic process

• Functional SLLN

• Note that while SLLN says that at each t, Yn(t) converges to mt, FSLLN says that entire sample paths of the sequences of stochastic processes Yn converge to the non-random process mt.

Fluid limit for the non-stationary queue

Theorem: If

then,

where Y is the stochastic fluid process with environment process E, deterministic flow rate ri = i - i in state i and initial content Y(0)=y.

Example: MMPP/M/1 queue

• Take the reference environment process, E, to be the following 2-state continuous time Markov chain

• In state H the queue behaves like an M/M/1 with service rate and arrival rate H (H > )

• In state L the queue behaves like an M/M/1 with service rate and arrival rate L(L < )

• Fluid limit:

n=10

n=100

n=1000

Problems with fluid limits

Problems with fluid limits

Functional Central Limit theorem

• Define the ‘centered’ partial sums of Xi as

• Central Limit Theorem

• Define the continuous time process

Question: How does Zn(t) behave as n increases?

n=100

n=1000

n=10000

n=1,000,000

Functional Central Limit theorem

• FCLT (Donsker’s Theorem)

where B(t) is the standard Brownian motion (with drift coefficient 0 and diffusion coefficient 1)

• Brownian motion with drift coefficient and diffusion coefficient 2 is a real valued stochastic process with stationary and independent increments having continuous sample paths where

Functional Central Limit theorem

• While CLT says that for any t,

FCLT also shows that Zn(t) converges to an (a.s.) continuous stochastic process with independent increments.

• Note that just as CLT is a refinement of the SLLN, the FCLT is a refinement of the FSLLN and hence is more accurate.

Diffusion limit for the non-stationary queue

• Theorem: If

then

where Z is a zero-drift Brownian motion with diffusion coefficient 2

z depending on the limiting fluid process, Y, and environment process, E, as follows– If Y(t)=0, then 2

z = 0

– If Y(t)>0 and E(t)=I, then 2z = i

3 i2 + i

3Si2

Diffusion limit for the non-stationary queue

• Proof:

Lemma: Let Xi be a sequence of positive random variables. Define

Let

denote the counting process with Xi as the interarrival times. Then,

Diffusion limit for the non-stationary queue

• Proof contd.Using the lemma on last slide, the counting process of arrivals, VA

n(t) in environment i converges to

Similarly, the counting process for service completions converges to

Taking the difference of the above Brownian motions gives the

diffusion limit.

Some implications of fluid and diffusion limits

1. The fluid limits only depend on the means of the service and arrival processes. Therefore, the variability of the environment process affects the queues more than the variability of the arrival and service processes within each environment state.

2. The limiting distribution does not depends on moments higher than the second moments of arrival and service processes.

3. The fluid and diffusion limits still hold when the arrival and service processes are not i.i.d but weakly dependent. This is a consequence of the fact that FSLLN and FCLT hold under much weaker conditions.

Conclusions

• Fluid and Diffusion limits are powerful tools that produce asymptotically exact distributions by appropriately scaling time and/or space for otherwise intractable problems by stripping away unnecessary details of the statistical processes involved.

• Engineering Applications– Buffer Provisioning for Network Switches and Routers– Scheduling Service for Multiple Sources

Thank you!

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