Chapitre 4
Files d’attente pour la planification des
capacités
Système M/M/1
RESULTATS:
Probabilité stationnaire ou distribution:
pn = rn(1-r), n≥0
Où r = l/m est appelé le taux de trafic.
Ls
= nombre moyen de clients dans le système = l/(m-l)
Ws
= temps moyen passé dans le système = 1/(m-l)
Lq
= longueur moyenne de file d'attente = l2/(m-l)m
Wq
= temps d'attente moyen = l/(m-l)m
r
= taux d'utilisation du serveur = l/m
p0
= Taux d'oisiveté du serveur = 1 - l/m
P(n > k) = probabilité d'avoir plus de k clients = (l/m)k+1
2
master
Système M/M/c – Modèle Erlang C
Un système M/M/c est une file d'attente :
• composé de c serveurs identiques
• dont les arrivées forment un processus de POISSON
• la durée de service suit une distribution exponentielle.
Le processus N(t), le nombre de clients présents dans le système à la date t,
est un processus de naissance et de mort.
• Taux de l'événement "arrivée" : l.
• Taux de "fin de service" : N(t)m si N(t)  c et cm si N(t) > c.
Condition de stabilité: l< c m.
3
master
Système M/M/c – Modèle Erlang C
Probabilité stationnaire ou distribution:
a= l/m : offered load
r = l/cm = a/c: traffic intensity
pn = an/n! p0,  0 < n  c
n
p nc
p0
a
=   p c ,n  0
c 
-1
c
 c -1 a n

a
= 




n
!
c
!
1
r


 n=0

m
0
m
1
l
m
2
l
m
3
l
l
4
master
Système M/M/c – Modèle Erlang C
Ls
= nombre moyen de clients présents dans le système
= Lq + a
Ws
= temps moyen passé dans le système des clients
= Wq + 1/m
Lq
= longueur moyenne de file d'attente
=
r
1 - r 
2
pc
Wq
= temps d'attente moyen
= Lq / l
a
= nombre de serveurs occupés, a = a
5
master
Système M/M/c – Modèle Erlang C
C(c,a)
= probabilité de délai de prise en charge d’un client
= pc + pc+1 + ...
a
C  c, a  =
pc
1- r
c ! 1 - r 
=
c -1

n=0
wq
a
n

n!
a
c
:
E rlang C form ula
c !1 - r 
= random waiting time of a customer

 0,
wq = 

 EXP cm - l  ,
T
g(T)
c
w ith probability 1 - C  c , a 
w ith probability C  c , a 
= Waiting time target
= Service level
= P(wq ≤ T)
g T  = 1 - C  c, a  e
-  c m - l T
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master
M/M/c avec clients impatients –Erlang B
• Le système est similaire au système M/M/c à l’exception des clients
perdus arrivés lorsque tous les serveurs sont occupés.
m
m
1
2
0
l
l
7
master
M/M/c avec clients impatients –Erlang B
Probabilité stationnaire ou distribution:
a= l/m : offered load
r = l/cm : traffic intensity
pn = an/n! p0,  0 < n  c
p0
 c an 
= 



n
!
 n=0

-1
8
master
M/M/c avec clients impatients –Erlang B
Erlag loss function or Erlang B formula
= Pourcentage des clients perdus ou overflow probability
B  c, a  = p c =
a
c
c
c!
 n=0 a
n
n!
Charge réelle des serveurs

a 1 - B  c, a 

9
master
M/M/c avec clients impatients –Erlang B
Normal approximation for staffing Erlang Loss systems
Condition: high offered load (a > 4) and high targeted service level
N(t) = number of patients : approximately normally distributed
E[N(t)]  a
In M/M/∞ system, N(t) =d POISSON(a), i.e. E[N(t)] = a, Var[N(t)] = a
Square-Root-Staffing-Formula for a delay probability a
c=a
a
 N -a
c-a
P  D elay  = a = P N  t   c = P 

 = 1-  
a
a





Where  is the cdf of the standard normal distribution
10
master
Computation issues of Erlang B and C formula
B  c, a  =
a

c
a
c!
c
n
a
n=0
C  c, a  =
n!
c ! 1 - r 
c -1

n=0
R  c, a  = 1 / B  c, a :
R  c, a  = 1 
B  c, a  =
C  c, a  =
c
a
n
n!

a
c
c ! 1 - r 
the reciprocal
cR  c - 1, a 
a
r B  c - 1, a 
1  r B  c - 1, a 
1 - r 
1 B  c - 1, a    r 1 - r 
r
B  0, a  = 1
C  0, a  = 1
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master
Approche file d’attente - Staffing the number of nurses
A hospital is exploring the level of staffing needed for a booth in the
local mall, where they would test and provide information on the
diabetes. Previous experience has shown that, on average, every 6.67
minutes a new person approaches the booth. A nurse can complete
testing and answering questions, on average, in twelve minutes.
Assuming s = 2, 3, 4 nurses, a hourly cost of 40€ per nurse and a
customer waiting cost of 75€ per hour in the system.
Determine the following: patient arrival rate, service rate, overall
system utilisation, nb of patients in the system (Ls), the average queue
length (Lq), average time spent in the system (Ws), average waiting
time (Wq), probability of no patient, probability of waiting, total system
costs.
21
Approche file d’attente - Staffing the number of nurses
Patien arrival rate
service rate
Overall system utilisation
L (system)
Lq
w (system) - in hours
Wq - in hours
no patient probability (idle)
patient waiting proba
Total system cost € per hour
9
5
90%
9,47
7,67
1,05
0,85
0,05%
85,26%
790
9
5
60%
2,33
0,53
0,26
0,06
14,60%
35,50%
205
9
5
45%
1,91
0,11
0,21
0,01
16,16%
12,85%
303
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Approche file d’attente - number of beds
Target occupancy level
Consider obsterics units in hospitals. Obsterics is generally operated
independently of other services, so its capacity needs can be determined
without regard to other services. It is also one for which the use of a
standard M/M/s queueing model is quite good. Most obsterics patients
are unscheduled and the assumption of Poisson arrivals has been shown
to be a ggod one in studies of unscheduled hospital admissions. In
addition, the coefficient of variation (CV) of the length of stay (LOS),
which is defined as the ratio of the standard deviation to the mean, is
typically very close to 1 satisfying the service time assumption of the
M/M/s model.
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Approche file d’attente - number of beds
Since obsterics patients are considered emergent, the American College
of Obsterics and Gynecology (ACOG) recommends that occupancy levels
of obsterics units not exceeding 75%. Many hospitals have obsterics
units operating below this level. However, some have eliminated beds to
reduce « excess » capacity and costs and 20% of NY hospitals had
obsterics units that would be considered over-utilized by this standard.
Assuming the target occupancy level of 75%, what is the probability of
delay for lack of beds for a hospital with s = 10, 20, 40, 60, 80, 100,
150, 200 beds.
Lesson :
For the same occupancy level, the probability of delay decreases with the
size of the service.
24
Approche file d’attente - number of beds
Evaluation of capacity based on a delay target leads to very important
conclusion. Though there is no standard delay target, it has been
suggested that the probability of delay for an obsterics bed should not
exceed 1%.
What is the size of an obsterics unit (nb of beds) necessary to achieve a
probability of delay not exceeding 1% while keeping the target
occupancy level of 60%, 70%, 75%, 80%, 85%?
Lesson :
Achieving high occupancy level while having small probability of delay is
only possible for obsterics unit of large hospitals.
Capacity cut should be made with clear understanding of the impact.
Simple and naive analysis based on average could lead to bad decisions.
25
Approche file d’attente - number of beds
Impact of seasonality
Consider an obsterics unit with 56 beds which experiences a significant
degree of seasonality with occupancy level varying from a low of 68%
in January to about 88% in July.
What is the probability of delay in January and in July?
If, as is likely, there are several days when actual arrivals exceed the
month average by 10%, what is the probability of delay for these days
in July?
Lesson :
Capacity planning should not be based only on the yearly average. Extra
bed capacity should be planned for predictable demand increase during
peak times.
26
Approche file d’attente - number of beds
Impact of clinical organisation
Consider the possiblity of combining cardiac and thoracic surgery patients as
thoracic patients are relatively few and require similar nursing skills as cardiac
patients.
The average arrival rate of cardiac patients is 1,91 bed requests per day and
that of thoracic patients is 0,42. No additional information is available on the
arrival pattern and we assume Poisson arrivals. The average LOS (Length Of
Stay) is 7,7 days for cardiac patients and 3,8 days for thoracic patients.
What is the number of beds for cardiac patients and thoracic patients in order
to have average patient waiting time for a bed E(D) not exceeding 0,5, 1, 2, 3
days? What is the number of beds if all patients are treated in the same
nursing unit?
Delay in this case measures the time a patient coming out of surgery spends
waiting in a recovery unit or ICU until a bed in the nursing unit is available.
Long delays cause backups in operating rooms/emergency rooms, surgery
cancellation and ambulance diversion.
27
Approche file d’attente - number of beds
Lesson :
Personal and equipment flexibility and service pooling can achieve higher
occupancy level and reduction of beds.
However, priority given to one patient group could significantly degrade
the waiting time of other patients if all treated in the same nursing unit.
27
Staffing ED with Erlang loss systems
Goal: meeting loss probability target (<5%, 1%) for stationary arrival
rate or dynamic arrival rate (Erlang B)
27
Staffing ED with Erlang C systems
Goal: meeting waiting time target for stationary arrival rate or dynamic
arrival rate
1: less than 20 minutes for at least 80% of patients
2: less than 1h for at least 95% of patients
27