Abstract

This paper introduces a discrete-time model which captures the essential protocol features of the congestion control mechanism used by the TCP Reno protocol, subject to interference from other sources. Under this model, a single target session is modeled according to the TCP Reno mechanism, including slow start, congestion avoidance, fast retransmit and fast recovery. At the same time, other sources are modeled as a background process using a discrete batch Markov arrival process (D-BMAP). The D-BMAP process has been modified such that the transitions between the phases are dependent on the number of lost packets from the background process. This introduces a feedback process, which can be used to model an aggregation of TCP sources. In order to capture all the TCP Reno protocol features, two levels of Markov process modeling are used: a microscopic level, at the packet transmission time boundaries, and a macroscopic one, at the start of the new transmission windows. In addition, it is shown how the model can be extended to model networks with RED-based routers.Several performance measures are derived, and numerical examples which demonstrate the protocol features are presented.

Keywords: Transmission control protocol (TCP); TCP Reno; Modeling; Markov chains; Discrete Batch Markov Arrival Process (D-BMAP); Random early detection (RED)

Nomenclature

a_n

average queue size when the n th packet arrives

B

router’s buffer size

number of D-BMAP phases

P(accept|l,ρ)

probability of accepting a packet given that l packets have been accepted, and the state is ρ

P(discard|l,ρ)

probability of discarding a packet given that l packets have been accepted, and the state is ρ

q_n

instantaneous queue size when the n th packet arrives

transition probability matrix between slot boundaries A subscript of s, i, or l is added if the event of success, idle of loss from the target is jointly considered

transition probability matrix with exactly j packets being accepted from the target session in the router’s queue within k slots

W_max

maximum (receiver advertized) window size

transition probability matrix between embedding points. A subscript i is added () if the initial window size is i. The subscript is i+ or i−, if success or loss from the target source is encountered within the interval

similar to , but joint with the event of loss that is detected through timeout. The window size may not increase, therefore

similar to , but when loss is detected through three duplicate acknowledgments

similar ot , and loss is detected during the same window as the loss

similar ot , and loss is detected during the window following the window of the loss

W(q)

block matrix row of when Ψ=q. Similar notation is used for , , and

transition probability matrix between two successive window evolution embedding points, started with a window size of k, when the target source is in the slow start phase

x(i,m|j,n,l,ρ)

P (within a slot, the buffer accepts i packets from the target session, given that j packets have already been accepted from the same session, and also accept m out of n packets from the background session, given l packets have been accepted, and the system state at the beginning of the slot is ρ)

y(0,m|0,n,l,ρ)

similar to x(), except that no packets are generated or accepted from the target session

z(i,m|j,n,l,ρ)

P (within a slot, i packets are discarded from the target session, given j have already been discarded, and also m packets are accepted out of n packets from the background session, given l have been accepted, and given the system state at the beginning of the slot is ρ)

Greek letters

α_f(n)

Pr(n packet arrivals | current phase is f)

β_ff^′(l)

Pr(next phase is f^′| current phase is f, and l packets have been discarded)

χ

timeout interval

η

a scaling factor used for scaling the difference between q_n and a_n−1

the D-BMAP phase at the slot boundaries

θ

the router’s queue size at the slot boundaries

τ

round trip propagation delay

ω

the window size of the target session at the slot boundaries

ψ

the system state at the slot boundaries (microscopic model)

Φ

the D-BMAP phase at the embedding points

Θ

the router’s queue size at the embedding points

Ω

the window size of the target session at the embedding points

Ψ

the system state at the embedding points (macroscopic model)

Δ

a function of the difference between the average and instantaneous queue sizes

Article Outline

Nomenclature

1. Introduction

2. Background

2.1. The TCP congestion control mechanism

2.2. TCP Reno

2.3. RED

2.4. The D-BMAP process

2.5. Related work

3. The model

4. Transition probabilities

4.1. Microscopic level analysis

4.1.1. Microscopic level analysis: no loss from the target session

4.1.2. Microscopic level analysis: loss from the target session

4.2. Macroscopic level analysis

4.2.1. Macroscopic level analysis: an interval without loss

4.2.2. Macroscopic level analysis: an interval with loss

Detection of loss through duplicate acknowledgments

Detection of loss through time-out

4.3. Transition probability matrix

5. Model extension: RED-based routers

6. Performance measures

7. Numerical examples

7.1. Model verification

7.2. Effect of system paramters

7.2.1. Effect of background traffic

7.2.2. Effect of buffer size

7.2.3. Effect of window size

7.3. Window size distribution

7.4. Packet discarding distribution

7.5. Summary of results

8. Conclusions

Acknowledgements

References

Vitae

This research started when the author was with Kuwait University, and was continued at Iowa State University under partial support by a Carver Trust Grant from Iowa State University.