SME
Computer Networks · Problem type
Queueing delay
Use this for “Packets reach a router at 800 a second and it serves 1000 a second. Find the average delay and number queued.”
This is an example. Type your own values to start a new problem.
T (ms)
=
1
μ (packets/s)
−
λ (packets/s)
N
=
ρ
1
−
Wq
Wait in queue
worked out (not typed)
Formulas
ρ
=
λ
μ
0.8
=
800
1,000
T
=
1
μ
−
λ
0.005
=
1
1,000 − 800
N
=
ρ
1 −
ρ
4
=
0.8
1 − 0.8
Wq
=
ρ
×
T
0.004 = 0.8 × 0.005
Formulas are worked in s. The step-by-step shows the conversions.
Assumptions
•
Arrivals are random (Poisson) and service times exponential: one server, an unlimited buffer, first come first served.
•
Rates are packets a second (s⁻¹); the queue settles only when λ < μ.
•
The time in system T is the wait plus the service, and N counts the packet being served.
Step-by-step
WE KNOW
λ = 800 packets/s, μ = 1,000 packets/s
FIND
utilization (ρ), time in system (T), packets in system (N), wait in queue (Wq)
Step 1 · Find utilization (ρ)
Use ρ = λ ÷ μ
The share of time the server is busy: arrivals over what it can serve.
ρ
=
800
1,000
ρ
=
0.8
Step 2 · Find time in system (T)
Use T = 1 ÷ (μ − λ)
The
server
clears
the
backlog
at
the
spare
rate
μ
−
λ
,
so
a
packet
spends
1
over
it
in
the
system.
T
=
1
1,000 − 800
T
=
1
200
T
=
0.005
s
Step 3 · Find packets in system (N)
Use N = ρ ÷ (1 − ρ)
The
busy
share
over
the
idle
share;
Little’s
law
N
=
λT
gives
the
same
number.
N
=
0.8
1 − 0.8
N
=
0.8
0.2
N
=
4
Step 4 · Find wait in queue (Wq)
Use Wq = ρ × T
The
time
in
system
less
one
service
time,
1
μ
,
is
ρ
of
it:
the
wait
before
service
starts.
Wq = 0.8 × 0.005
Wq = 0.004 s
Step 5 · Convert the answers
Change each answer back to the unit you chose.
T = 0.005 s = 5 ms (1 s = 1000 ms)
Wq = 0.004 s = 4 ms (1 s = 1000 ms)
Check
Put the numbers back in. Both sides should match.
0.8
=
800
1,000
✓
0.005
=
1
1,000 − 800
✓
4
=
0.8
1 − 0.8
✓
0.004 = 0.8 × 0.005 ✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.
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