Performance analysis for high speed switches Lecture 6 The M/M/1 Queueing System The M/M/1 Queueing System The M/M/1 Queueing System consisits of a single queueing station with a single server. The name M/M/1 reflects standard queueing theory nomenclature whereby:
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Between the basic quantities,
N = Average number of customers in the system
T = Average customer time in the system
NQ = Average number of customers waiting in queue
W = Average customer waiting time in queue
and using Little’s Theorem,
and are mutually independent.
where we generically denote by o(δ) a function of δsuch that
for the interarrival and service times and ,respectively
where δis a small positive number. We denote
Nk= Number of customers in the system at time
These equations can also be written as
and finally, using , we have
Using , this becomes
Average Number in the system N
need not be equal to the corresponding unconditional steady-state probabilities,
and we obtain
where W is the expected customer waiting time in queue and
we mean that the customer j is already being serve
when i arrives, is the remaining time until customer
j’s service time is complete. If no customer is in
service(i.e., the system is empty when i arrives), then
customer upon arrival
R = Mean residual time, define as
Residual Service time γ（τ）
and by substitution in the waiting time formula, we obtain
where is the utilization factor; so finally,
where M(t) is the number of service completions within[0,t], and Xiis the service time of the ith customer.
and assuming the limits below exist, we obtain
Output Queueing -- "We wait at the destination (output) together"Input Queueing versus Output Queueing
p together" = load
Analysis of Output Queueing
: the number of packets in the tagged queue at the end of the time slot m
Using a standard approach in queueing analysis
The mean queue size for an M/D/1 queue
The mean stead-state queue size
2The State transition diagram for the output queue size
The time slots that packet must wait while packets that arrived in earlier time slots are transmitted
The time slots that packet must wait additionally until it is randomly selected out of the packet arrivals in the time slot m
The mean waiting time for an M/D/1 queue
Internally Nonblocking Switch
cannot access output 2 because it is blocked by the first packet
0= Pr[ carry a packet ]
Pr[ carry a packet ] =p
for large N
For p=1, 0= 0.632
Fictitious Output Queues formed by HOL packets
Internally Nonblocking Switch
* : the maximum throughput with input queueing
Throughout of Input-Buffered Switch together"
= # packets at start of time slot m.
= # packets arriving at start of time slot m.
e.g. Fictitious Queue together" i
time slot m
time slot m-1
Input Queue together"
For finite buffer size, if p0 > p* = 0.586 at least (p0 - p*)/ p0 fraction of packets are dropped.
Must keep p0 < p*
Meaning of Saturation Throughput
p0 = p = throughput
Time spent in HOL are independent for successive packets when N is large
Service times at different fictitious queues are independent
Arrivals here are considered as arrivals in intervals i-2
Arrivals here are considered as arrivals in intervals i-1
mi =2 prior arrivals
Arrival of the packet of focus. One simultaneous arrival to be served before the packet; L=1.
Departure of packet of focus.
-- Packet arrival in interval i.
-- packet departure in interval i+1.
-- number of arrivals
Different contention-resolution policies have different waiting time versus load relationships, but a common maximum load at which waiting time goes to infinity.