Application of QueuingTheory to Data

Communications Systems

Queuing Theory for Data:End-to-End Delays

Three Useful Models of System Delay:

1. Applicable end-to-end queuing delay = Taq or tot. queuing delay experienced by a block

2. Block turnaround time = Tta or tot. turn-around time experienced by a block

3. “Bid-to-start of message” = BST or the totaldelay measured from bid to the SOM.

–Time

T

T

T

T

R

R

R

R

DTE

PS2

PS1

Host

BST

K1

K2

A Time Chart for Modeling Response Time

HTT

Queuing Theory for Data:Nodal Queuing Times

PS Node: E ( Tq ) =E ( Tq1 )+E ( Tq2 )

Input Process 1:(assume M/D/1 queue)

E ( Tq1 ) = [Ts1 (2- /[2 (1-)]

Output Process 2:(assume M/D/1 queue)

E ( Tq2 ) = [Ts2 (2- /[2 (1-)]

where Tqi , Tsi and i represent ith server(i=1, 2) i.e. PS-CPU and a single outputline.

Queuing Theory for DATA:Nodal Queuing Times

Fast Packet Switch (FPS) & Frame-Relay:

E( Tq )= 0 .... pass through packet

E( Tq )=[ s ...local packet

where andare netlink utilizationscaused by pass through and local trafficssuch that + =

FPS

SNA Data Nets: Avg. BTS Delay

BTS=E(D)= (W/2)[1-/N]/(1 -)+Tmm/(1 -)

This is equivalent to Equation 4.21 of TEXTwhere

= Avg. utilization of the shared netlink

= Nm..... is msg. arrival rate/contrllr

Tmm=EMD=exp. msg. duration=m

N=No. of controllers per netlink

W=walk time (Idle poll of N controllers).

SNA Networks: Avg. Walk Time

Avg. Walk Time for SNA with host polling:

W=4N(Tnp) +2N(Tpm)+2N(Tm)+N+1)/2

Avg. Walk Time for SNA with hub polling:

W=4N(Tnp) +2N(Tpm)+2N(Tm)+where Tnp is nodal msg. processing time, Tpm is

poll message transmit time, Tm is modem timeand is propagation time on the entirenetlink consisting of N controllers.

Note: Above 4 Eqtns. are from M. Schwartz’s Text.

Queuing Theory for DATA:Block Turnaround Times

SNA Application: N controllers on netlink.

Tta = 4NTnp+ +2N(Tm+2Tpm)

FPS Application:N FPSs sharing a netlink.

Tta 4Tnp+ + 2N(Tm + 2Tmm)

where Tnp=nodal processing time, Tm ismodem time, is the 2-way propagationtime on the netlink, Tpm is poll messagetransmit time and Tmm is packet/blocktransmit time. FPS uses nodal bypass.

X.25 PS System:Block TA Time

• Tta=4NTnp +N+1)/2 +N(Tm+Tmm)

where

Tta= block turnaround time

Tnp=nodal processing time

Tm=modem time

Tmm=packet/block time

= 2-way propagation time

Note: Tnp includes ACK and NAK times

LANs:Shared Media Utilization

Considering two extreme cases when (A)only one node is active and (B) N nodes areactive, one can compute the maximummean throughput rate in bits per second:

1. Token Ring (TR) LAN as employed inoffice environements:

Rp= Lm /[Tmsg +Tp +N*Tint]bps....Case A

Rp= Lm /[Tmsg +Tint + Tp/N]bps ....Case B

LANs:Shared Media Utilization

Considering two extreme cases when (A)only one node is active and (B) N nodes areactive, one can compute the maximum meanthroughput rate in bits per second:

2. Token Bus LAN as employed inmanufacturing environments:

Rp= Lm /[Tmsg +N*(Tint+Tp]bps....Case A

Rp= Lm /[Tmsg +Tint + Tp]bps ..... Case B

LANs Shared Media Utilization

Considering two extreme cases when (A) onlyone node is active and (B) N nodes areactive, one can compute the maximum meanthroughput rate in bits per second:

3. CSMA/CD LAN as employed in offices:

Rp = Lm /[Tmsg +Tifg] .... Case A

Rp = Lm /[Tmsg +Tifg +(2e-1)( Ts+ Tj)] ..... Case B

where Tifg ,Ts and Tj are interframe gap, slotand jam times respectively

LANs Shared Media Utilization

Assumptions:

1. N =100

2. Only onenode active

3. Packet/Msgsize=2000 bits

10

5

02515 201050

20

25

15

Rp

Media capacity (Mbps)

(Mbps)

TR

TBus

CSMA/CD

LANs Shared Media Utilization

10

5

02515 201050

20

25

15

Rp

Media capacity (Mbps)

(Mbps)

Assumptions:

1. N =100

2. All nodesare active

3. Packet/Msgsize=2000 bits CSMA/CD

TR

TBus

LANs Performance (Contd.)

1. LAN performance for any given operatingenvironment is almost impossible to analyzeusing analytical simulation.

2. Simplified analytical models can be createdand when combined with a spreadsheet canprovide meaningful answers for simple set ofoperating environs.

3. Computer simulation can be employedjudiciously to derive useful analytical modelsaccording to the scheme discussed earlier.

50*Ts

0. 5

25

00 1

Aloha U

AlohaS CSMA/CD

TR X.25

FPS

* X.25 & FPS have 10 link-nodes* TR for*CSMA/CD for = 0.1 and 0.05

= Tp/Ts

where Tp =prop. time

Ts=slot. time

= (Tp+N*Tint)/Ts

Legend:

AlohaU for Unslotted,

AlohaS for Slotted,

TR =Token Ring

FPS =Fast PS

LANs/WANs Performance Vs.

0.1 .05

shared media load

1-way delay

Class Assignments

1. Redraw Figure 4.4 for=0.75

2. Redraw Figure 4.5 for=0.75