Chapitre 4 Files dattente pour la planification des capacités

Chapitre 4Files d’attente pour la planification des

capacités

master GI2007

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Système M/M/1Système M/M/1

RESULTATS:RESULTATS:

Probabilité stationnaire ou distribution:Probabilité stationnaire ou distribution:

nn = = nn(1-(1-), ), n≥0n≥0

Où Où = = // est appelé le taux de trafic. est appelé le taux de trafic.

Ls Ls = nombre moyen de clients dans le système = = nombre moyen de clients dans le système = /(/())

WsWs = temps moyen passé dans le système = 1/(= temps moyen passé dans le système = 1/())

LqLq = longueur moyenne de file d'attente = = longueur moyenne de file d'attente = 22/(/())

WqWq = temps d'attente moyen = = temps d'attente moyen = /(/())

= taux d'utilisation du serveur = = taux d'utilisation du serveur = //

00 = Taux d'oisiveté du serveur = 1 - = Taux d'oisiveté du serveur = 1 - //

P(n > k)P(n > k) = probabilité d'avoir plus de k clients = (= probabilité d'avoir plus de k clients = (//))k+1k+1

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Système M/M/c – Modèle Erlang C Système M/M/c – Modèle Erlang C

Un système M/M/c est une file d'attente :Un système M/M/c est une file d'attente :

• composé de c serveurs identiquescomposé de c serveurs identiques

• dont les arrivées forment un processus de POISSONdont les arrivées forment un processus de POISSON

• la durée de service suit une distribution exponentielle. la durée de service suit une distribution exponentielle.

Le processus N(t), le nombre de clients présents dans le système à Le processus N(t), le nombre de clients présents dans le système à la date t, est un processus de naissance et de mort. la date t, est un processus de naissance et de mort.

• Taux de l'événement "arrivée" : Taux de l'événement "arrivée" : ..

• Taux de "fin de service" : N(t)Taux de "fin de service" : N(t) si N(t) si N(t) c et c c et c si N(t) > c. si N(t) > c.

Condition de stabilité: Condition de stabilité: < c < c ..

master GI2007

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Système M/M/c – Modèle Erlang CSystème M/M/c – Modèle Erlang C

Probabilité stationnaire ou distribution:Probabilité stationnaire ou distribution:aoffered loadctraffic intensityn = an/n! 0, 0 < n c

0 1 2 3

11

00 ! ! 1

n cc

n

a a

n c

, 0n

n c ca

nc

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Ls = nombre moyen de clients présents dans le système= Lq + a

Ws = temps moyen passé dans le système des clients= Wq + 1/

Lq = longueur moyenne de file d'attente=

Wq = temps d'attente moyen= Lq /

= nombre de serveurs occupés, = a

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c

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C(c,a) = probabilité de délai de prise en charge d’un client= c + c+1 + ...

wq = random waiting time of a customer

T = Waiting time target(T) = Service level

= P(wq ≤ T)

1

0

! 1, : Erlang C formula

1

! ! 1

c

cn cc

n

a

cC c a

a a

n c

0, with probability 1 ,

, with probability ,q

C c aw

EXP c C c a

1 , c TT C c a e

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M/M/c avec clients impatients M/M/c avec clients impatients –Erlang B–Erlang B

• Le système est similaire au système M/M/c à l’exception des Le système est similaire au système M/M/c à l’exception des clients perdus arrivés lorsque tous les serveurs sont occupés.clients perdus arrivés lorsque tous les serveurs sont occupés.

0 1 2

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Probabilité stationnaire ou distribution:Probabilité stationnaire ou distribution:aoffered loadctraffic intensityn = an/n! 0, 0 < n c

1

00 !

nc

n

a

n


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Erlag loss function or Erlang B formula = Pourcentage des clients perdus ou overflow probability

Charge réelle des serveurs

0

!,

!

c

c c nn

a cB c a

a n

1 ,a B c a


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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

c a a


1N a c a

P Delay P N t c Pa a

Where is the cdf of the standard normal distribution

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0

!,

!

c

c nn

a cB c a

a n

Computation issues of Computation issues of Erlang B and C formulaErlang B and C formula

1

0

! 1,

! ! 1

c

n cc

n

a

cC c a

a a

n c

, 1 / , : the reciprocalR c a B c a

1,, 1

cR c aR c a

a

1,

1 1, 1C c a

B c a

1,,

1 1,

B c aB c a

B c a

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Approche file d’attente - Staffing the number of nursesApproche file d’attente - Staffing the number of nurses

A hospital is exploring the level of staffing needed for a booth in A hospital is exploring the level of staffing needed for a booth in the local mall, where they would test and provide information the local mall, where they would test and provide information on the diabetes. Previous experience has shown that, on on the diabetes. Previous experience has shown that, on average, every 6.67 minutes a new person approaches the average, every 6.67 minutes a new person approaches the booth. A nurse can complete testing and answering questions, booth. A nurse can complete testing and answering questions, on average, in twelve minutes.on average, in twelve minutes.

Assuming s = 2, 3, 4 nurses, a hourly cost of 40€ per nurse and 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. a customer waiting cost of 75€ per hour in the system.

Determine the following: patient arrival rate, service rate, Determine the following: patient arrival rate, service rate, overall system utilisation, nb of patients in the system (Ls), the overall system utilisation, nb of patients in the system (Ls), the average queue length (Lq), average time spent in the system average queue length (Lq), average time spent in the system (Ws), average waiting time (Wq), probability of no patient, (Ws), average waiting time (Wq), probability of no patient, probability of waiting, total system costs.probability of waiting, total system costs.

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Patien arrival rate 9 9 9service rate 5 5 5Overall system utilisation 90% 60% 45%L (system) 9,47 2,33 1,91Lq 7,67 0,53 0,11w (system) - in hours 1,05 0,26 0,21Wq - in hours 0,85 0,06 0,01no patient probability (idle) 0,05% 14,60% 16,16%patient waiting proba 85,26% 35,50% 12,85%Total system cost € per hour 790 205 303

Approche file d’attente - Staffing the number of nursesApproche file d’attente - Staffing the number of nurses

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Target occupancy levelTarget occupancy level

Consider obsterics units in hospitals. Obsterics is generally Consider obsterics units in hospitals. Obsterics is generally operated independently of other services, so its capacity needs operated independently of other services, so its capacity needs can be determined without regard to other services. It is also 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 one for which the use of a standard M/M/s queueing model is quite good. Most obsterics patients are unscheduled and the quite good. Most obsterics patients are unscheduled and the assumption of Poisson arrivals has been shown to be a ggod one assumption of Poisson arrivals has been shown to be a ggod one in studies of unscheduled hospital admissions. In addition, the in studies of unscheduled hospital admissions. In addition, the coefficient of variation (CV) of the length of stay (LOS), which is coefficient of variation (CV) of the length of stay (LOS), which is defined as the ratio of the standard deviation to the mean, is defined as the ratio of the standard deviation to the mean, is typically very close to 1 satisfying the service time assumption typically very close to 1 satisfying the service time assumption of the M/M/s model.of the M/M/s model.

Approche file d’attente - number of bedsApproche file d’attente - number of beds

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Since obsterics patients are considered emergent, the American Since obsterics patients are considered emergent, the American College of Obsterics and Gynecology (ACOG) recommends that College of Obsterics and Gynecology (ACOG) recommends that occupancy levels of obsterics units not exceeding 75%. Many occupancy levels of obsterics units not exceeding 75%. Many hospitals have obsterics units operating below this level. hospitals have obsterics units operating below this level. However, some have eliminated beds to reduce « excess » However, some have eliminated beds to reduce « excess » capacity and costs and 20% of NY hospitals had obsterics units capacity and costs and 20% of NY hospitals had obsterics units that would be considered over-utilized by this standard.that would be considered over-utilized by this standard.

Assuming the target occupancy level of 75%, what is the Assuming the target occupancy level of 75%, what is the probability of delay for lack of beds for a hospital with s = 10, probability of delay for lack of beds for a hospital with s = 10, 20, 40, 60, 80, 100, 150, 200 beds.20, 40, 60, 80, 100, 150, 200 beds.

Lesson : Lesson :

For the same occupancy level, the probability of delay decreases For the same occupancy level, the probability of delay decreases with the size of the service.with the size of the service.


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Evaluation of capacity based on a delay target leads to very Evaluation of capacity based on a delay target leads to very important conclusion. Though there is no standard delay target, important conclusion. Though there is no standard delay target, it has been suggested that the probability of delay for an it has been suggested that the probability of delay for an obsterics bed should not exceed 1%.obsterics bed should not exceed 1%.

What is the size of an obsterics unit (nb of beds) necessary to What is the size of an obsterics unit (nb of beds) necessary to achieve a probability of delay not exceeding 1% while keeping achieve a probability of delay not exceeding 1% while keeping the target occupancy level of 60%, 70%, 75%, 80%, 85%?the target occupancy level of 60%, 70%, 75%, 80%, 85%?

Lesson : Lesson :

Achieving high occupancy level while having small probability of Achieving high occupancy level while having small probability of delay is only possible for obsterics unit of large hospitals.delay is only possible for obsterics unit of large hospitals.

Capacity cut should be made with clear understanding of the Capacity cut should be made with clear understanding of the impact. Simple and naive analysis based on average could lead to impact. Simple and naive analysis based on average could lead to

bad decisions. bad decisions.


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Impact of seasonalityImpact of seasonality

Consider an obsterics unit with 56 beds which experiences a Consider an obsterics unit with 56 beds which experiences a significant degree of seasonality with occupancy level varying significant degree of seasonality with occupancy level varying from a low of 68% in January to about 88% in July.from a low of 68% in January to about 88% in July.

What is the probability of delay in January and in July?What is the probability of delay in January and in July?

If, as is likely, there are several days when actual arrivals If, as is likely, there are several days when actual arrivals exceed the month average by 10%, what is the probability of exceed the month average by 10%, what is the probability of delay for these days in July?delay for these days in July?

Lesson : Lesson :

Capacity planning should not be based only on the yearly average. Capacity planning should not be based only on the yearly average. Extra bed capacity should be planned for predictable demand Extra bed capacity should be planned for predictable demand

increase during peak times.increase during peak times.


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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.


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Lesson : Lesson :

Personal and equipment flexibility and service pooling can Personal and equipment flexibility and service pooling can achieve higher occupancy level and reduction of beds.achieve higher occupancy level and reduction of beds.

However, priority given to one patient group could significantly However, priority given to one patient group could significantly degrade the waiting time of other patients if all treated in the degrade the waiting time of other patients if all treated in the

same nursing unit.same nursing unit.


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Goal: meeting loss probability target for stationary arrival rate or Goal: meeting loss probability target for stationary arrival rate or dynamic arrival ratedynamic arrival rate

Staffing ED with Erlang loss systemsStaffing ED with Erlang loss systems

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Goal: meeting waiting time target for stationary arrival rate or Goal: meeting waiting time target for stationary arrival rate or dynamic arrival ratedynamic arrival rate

Staffing ED with Erlang C systemsStaffing ED with Erlang C systems

Documents

Chapitre 4 Files dattente pour la planification des capacités