COUNTING PROCESSES

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COUNTING PROCESSES

• Very important modelling concept
• e.g.
• Arrival of a product at a machine
• Arrival of emails to a server
• Arrival of calls to a call centre
• Arrival of a signal to a processor
• Described by a counting function N (t) for all t
  0
• Counts how often events occur over time
• Generally, A system that is characterized by
  customers contending for a resource, called a
  server, is a queuing system, or counting process.

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• N(t), t 0 a counting process
• Assume
• Arrivals occur one at a time
• Arrivals are completely random
• without rush or slack periods (memoryless)
• 3. All arrivals are independent
• Then counting process ? Poisson process
• So Poissonian
• Mean and variance ?t

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If first arrival at time A1, second one at time
A1A2 etc.. Then P(A1 t ) 1-e-?t i.e. A1
exponentially distributed with mean 1/? Also
all interarrival times A1, A2. exponentially
distributed with mean 1/? If independent and
memoryless we have a Poisson process Often
used to trade off server utilization with
customer satisfaction In terms of line lengths
and delays
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CHARACTERISTICS OF QUEUEING SYSTEMS Key
elements customers and servers very
generic Simplest system Population of
customers calling population finite or
infinite
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Example scenario A set of 5 machines that
process (cure) tires After t ? a machine
opens and a worker removes a tire and loads an
uncured tire and closes the machine Machines
customers who arrive at a rate dependent on the
distribution of the t ? Values Worker
server calling population 5 Arrival rate
expected number of arrivals in next time unit ?
If all 5 machines are closed arrival rate
maximum
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• Note
• Infinite populations arrival rates usually
  modelled as constant
• Finite populations other statistical models
  used
• SYSTEM CAPACITY
• Defined when number of customers in the waiting
  line or system is limited e.g. with space
  constraints
• Define arrival rate effective arrival rate
• (no. of arrivals)
  (no. of arrivals that enter system)

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Properties of the Poisson process Random
splitting A Poisson process of rate ? may have
2 event types (I and II) of probabilities (P)
and (1-P) N1(t) random variable for number of
type I events N2(t) random variable for number
of type II events Such that N(t) N1(t)
N2(t) Then.
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Both are also Poisson processes with rates ?P
and ?(1-P) Reverse is also true pooled
process e.g. 2 Poisson arrival streams of
rates ?1 10/hour and ?2 17/hour are combined
at a system entity and moved on Result
Poisson process of ?27/hour
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QUEUEING MODELS Queueing models most common
model in engineering
Something arrives at an entity, waits, then goes
through
e.g. server utilization, waiting times, delays,
starvation, bottlenecks etc. Equally important
in manufacturing, electrical and computer
systems etc. What happens if a system is full? -
Customers either form a queue or are turned away
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Arrival Processes Infinite populations
customers may arrive in a random manner,
individually or in groups (batches) - modelled
using probability distributions Most important
model the Poisson process An interarrival
time between customers n-1 and n An
exponentially distributed with mean of 1/? time
units (arrival rate is ? customers per time
unit) Number of arrivals in time interval t
N(t) is Poisson with a mean of ?t
customers Very reliable and easy to model
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Proven examples restaurants, banks, call
centres, machine failures etc.
Finite populations? Here customers are pending
or non-pending Returning to the tire
system.. Pending machine curing (tire not
ready yet) Non-pending instant the machine is
open demands service from the worker Also
define runtime Runtime for a customer time
from departure from the queue to their next
arrival to the queue
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Let A1(i), A2(i) etc successive runtimes of
customer i Let S1(i), S2(i) etc. corresponding
successive system times Then for our example
(machine 3 as an illustration)
In the total system A1 will be min A1(1),
A1(2), A1(3), A1(4), A1(5) Arrival rate is not
constant dependent on number of pending
customers
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Best example repair of a failed system
entity Here Customers system entity Runtime
time to failure Entity fails arrives at
queueing system (repair facility) until served
(repaired) Exponential, gamma, Weibull
distributions used to model distribution of
failure times which are assumed to be independent
(not always true e.g. age etc.)
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QUEUE BEHAVIOUR When in a queue a customer
can Wait until called Balk (leave if line is too
long) Renege (leave if line is moving too
slowly) Jockey (switch from line to line if
moving at different rates) Good example
passport line at an airport Queue can be dealt
with On a first-in-first-out basis
(FIFO) Last-in-first-out basis (LIFO) Service in
random order (SIRO) Shortest processing time
first (SPT) Service according to priority (PR)
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System structure Queueing system number of
service centres interconnecting queues each
service centre has c servers working in
parallel customer takes first available
server Parallel service centres
either single c0 multiple c1 unlimit
ed c8 Self-service facility unlimited
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EXAMPLE discount warehouse where customers
either serve themselves or wait for one of three
clerks. Then pay through a single
cashier 1 subsystem Queueing
issues? Splitting? Pooling?
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QUEUEING NOTATION Many diverse queueing systems
need comprehensive notation KENDALL
notation A/B/c/N/K system A interarrival time
distribution B service-time distribution c
number of parallel servers N system capacity K
size of calling population A, B M
(exponential or Markov), D (constant or
deterministic) G (arbitrary or general), Ek
(Erlang of order k) etc.
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Examples M/M/1/8/8 Single-server system that
has unlimited queue capacity infinite
population of potential arrivals. Interarrival
and service times are exponentially
distributed Also designated M/M/1 Tire curing
system? G/G/1/5/5 system Notation assumes a
FIFO queue discipline unless told otherwise
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• Long-term performance measures
• for queueing systems
• Queueing system waiting line service
  mechanism
• Queue waiting line only
• For a time-dependent system three phases
• Switching on warming up ramp-up beginning
• Steady-state long-term use
• Winding down switching off closure etc
• Queueing theory applies to steady state behaviour

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Pn steady-state probability of having n customers
in system Pn(t) probability of n customers in
system at time t ? arrival rate ?e effective
arrival rate µ service rate of one server ?
server utilization An interarrival time between
customers n-1 and n Sn service time of the nth
arriving customer Wn total time spent in system
by nth arriving customer WnQ total time spent in
the waiting line by customer n L(t) the number of
customers in system at time t LQ(t) the number of
customers in queue at time t L Long-run time
average number of customers in system LQ Long-run
time average number of customers in
queue w Long-run average time spent in system per
customer wQ Long-run average time spent in queue
per customer
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L Long-run time average number of customers in
system A system over time T Ti when i
customers are in the system Here where
is an estimator of L So
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w average time spent in the system per
customer As an estimate Where Wi time spent
in system by customer I and N number of
arrivals over the measured time period In steady
state conditions, as N ? 8, ? w Previous example
assume FIFO arrangement Customer 1 arrives at
t0 and leaves at t2 Customer 2 arrives at t3
and leaves at t8 Customer 3 arrives at t5 and
leaves at t10 Customer 4 arrives at t7 and
leaves at t14 Customer 5 arrives at t16 and
leaves at t20
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So As for time spent in the waiting line
Littles Law From our system N 5 arrivals, T
20 time units So arrival rate, ? 5/20 ¼
customer per time unit From before So -
Littles Law of conservation important!
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? server utilization proportion of time
between 0 and T that the server is busy Again,
at steady-state Very dependent on type of
system
G/G/1/8/8 queues Any single-server queue system
of arrival rate ? per time unit Average service
time E(S) 1/µ time units ? When busy
server works at rate of µ customers/time unit µ
service rate Littles Law is valid
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Actual number of customers in the server 0 or
1 (rest are in the queue) Our example
again.. Now L merely Combining into
Littles Law gives And is always less than 1 in
stable situations
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G/G/c/8/8 queues
Here - ? Ls /c ?/cµ Where Ls average
number of busy servers (0Lsc) Maximum
service rate cµ when all servers are busy To
be stable ? CASE STUDY Q1
Each clerk is busy 50 of the time On average
10 clerks are being used ? - what would be a good
number?
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D/D/1 example
Consider a doctor who schedules patients every 10
minutes and who spends Si minutes with the ith
patient where
Arrivals are deterministic (constant) with ?-1
10 E(A) Service time mean of E(Si) 9(0.9)
12(0.1) 9.3 minutes and variance V(Si)
E(Si2) E(Si)2 92(0.9) 122(0.1)
(9.3)2 0.81 mins So ? ?/µ E(S)/E(A)
9.3/10 0.93
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Steady-state prediction of Markovian
queues i.e. anything with an M in it. If
population infinite - arrivals assumed to
follow Poisson behaviour with ? arrivals per
time unit, and interarrival times that are
exponential with mean 1/? - service times
exponential or arbitrary (M or G) If queue
discipline FIFO A Markovian model results
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Main Markovian property memoryless i.e. state
of system is time independent A good analysis
especially for long-term behaviour i.e. steady
state These are mathematical models not
simulation models Difference? read up!
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M/G/1 system Interarrival times 1/? 1 server
service times of mean 1/µ and variance
s2 Under steady state we have
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CASE STUDY Q2
? 1.5 per hour 1/µ ½ hour s2 400mins2 ?
?/µ 0.75 Consistency of units! s2 1/9
hour2 At any instant in steady-state time,
2.375 machines are broken
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LQ also written as Analysis of this shows
that waiting lines (hence delays) are mostly
dependent on s2 variability of service times
has biggest effect on queues and delays

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CASE STUDY Q3
Highlights the critical nature of variability to
the system and also the importance interpreting
the various answers from the analysis
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M/M/1 system Service times exponential (mean
1/µ) ?variance s2 1/µ2 Good when standard
deviations approximate to the mean
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CASE STUDY Q4
Here, ? 2/hour µ 3/hour Utilization ? ?/µ
2/3 How many customers are in the shop at any
one time? Use P0 0.33 P1
0.22 P2 0.15 P3 0.1 P4 0.2 Also the
hairstylist is idle 33 of the time (1-?)
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Average number of customers Average time
spent in the system Average time spent in the
queue per customer Average number of
customers in the queue
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EFFECT OF UTILIZATION AND SERVICE VARIABILITY In
any M/G/1 queue.. line size reduced by
decreasing server utilization or service time
variability s2 Reduce ? by decreasing arrival
rate ? increasing service rate µ increasing
no. of servers c Regarding s2 can define the
coefficient of variation (cv) In exponential
case E(X) 1/µ and V(X) 1/µ2 ?cv1
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Interestingly for an M/G/1 queue
M/M/1 model correction factor
True for other parameters An M/G/1 queue is
basically a M/M/1 queue with extra variation
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Multiserver queue M/M/c Here the parameters
are complex
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P0 complex expression but often graphed
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Also L
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CASE STUDY Q5
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