Fila Pra Tudo
Autor: André Luiz Diniz | Publicado: 2011-09-30T13:13:00-07:00

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Um dos maiores transtornos experimentados pelo público que foi aos três primeiros dias do Rock In Rio foi o excesso de filas. Havia filas enormes para pegar o ônibus, entrar na Cidade do Rock, comprar comida e, principalmente, se divertir na montanha russa, tirolesa ou na roda gigante. Em um dos stands no Bob´s, a espera para comprar um lanche chegava a quase duas horas e havia filas separadas para os cinco caixas existentes. Fiquei imaginando o porquê de não adotarem o sistema de fila única, dispondo os caixas lado a lado. Este procedimento, difundido no país por volta da década de 1990, é hoje comum em muitos estabelecimentos, como agências bancárias, bilheterias de casas de espetáculos, guichês do metrô e até em alguns supermercados. 
Este post mostra que o sistema de fila única seria a maneira mais justa e conveniente de atender às pessoas que quisessem comprar seu lanche no Bob´s do Rock In Rio. Superadas eventuais dificuldades práticas para a sua implementação, principalmente em relação à disposição da fila e dos atendentes, são inegáveis os ganhos da fila única em relação à utilização de uma fila separada para cada caixa. Evita que atendentes fiquem ociosos1, preserva a justiça na ordem de atendimento (que segue rigorosamente a ordem de chegada de clientes, observando-se o sistema como um todo) e diminui a incerteza no tempo em que uma pessoa ficará na fila até ser atendida. 
A questão mais importante no estudo de um sistema de filas é a representação adequada tanto do tempo até a entrada de um novo cliente na fila (que fará com que ela aumente) como do tempo até a saída de um cliente que esteja sendo atendido (que fará com que a fila diminua). No primeiro caso, a dificuldade está em identificar o padrão segundo o qual as pessoas chegam para serem atendidas e no segundo caso a dificuldade é em prever a duração de cada atendimento. 
A Estatística tem papel fundamental na modelagem desses dois tempos, que, por serem incertos, são denominados variáveis aleatórias. Apesar de ser impossível prever o tempo de chegada de um cliente ou a duração de um atendimento, pode-se construir uma espécie de histograma para cada uma dessas variáveis, mostrando a gama de valores que elas podem assumir e suas respectivas frequências de ocorrência. Com base nestas informações, pode-se simular numericamente a evolução da fila ao longo do tempo (utilizando o Excel ou algum software de estatística) e traçar histogramas de frequências para o número de pessoas na fila e para o tempo de espera até o atendimento, os quais permitem avaliar o desempenho do sistema2. 
Para exemplificar esse tipo de análise, simulei o que poderia ocorrer a uma pessoa que entrasse na fila para comprar um lanche no Bob’s do Rock In Rio em determinado momento quando havia 5 caixas, cada um com uma fila de aproximadamente 40 pessoas. O tempo médio de atendimento por cliente era de cerca de 3 minutos, valor estimado com base nas 2 horas de espera na fila, em média3. 
Considerei dois tipos de configuração para este sistema: o atendimento em filas separadas, como estava ocorrendo na prática, e um atendimento hipotético em fila única, que seria outra alternativa de prestar o serviço ao público. Neste segundo caso, ao invés da última pessoa que chegou estar na 40ª posição da fila de um dos caixas, ela seria a 200ª pessoa em uma fila única para os 5 caixas. Como se percebe, o número de pessoas por caixa é o mesmo em ambas as configurações.
Ressalto que, como o processo é aleatório, não é possível calcular quanto tempo a pessoa ficará na fila, mas apenas obter uma relação dos possíveis valores de tempo de espera e suas respectivas probabilidades. Veja a diferença entre os resultados dos sistemas de filas separadas e de fila única, obtidos a partir de 1000 simulações para cada uma dessas configurações.
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 <![endif]-->  Histograma de freqüências do tempo de espera na fila, nas duas configurações analisadas.
Apesar de o tempo médio em ambos os casos ser de 2 horas (120 minutos), há uma variação muito maior entre os possíveis tempos de espera no sistema de filas separadas. Observe a faixa de 15 minutos para mais ou para menos em relação ao tempo médio (ou seja, um tempo de espera entre 105 e 135 minutos): enquanto no sistema de filas separadas em aproximadamente 25,8% das vezes o tempo de espera ficou fora dessa faixa, no sistema de fila única isso só ocorreu em 1,4% das simulações.
Portanto, observa-se que, no sistema de filas separadas, há uma chance maior de esperar menos tempo do que a média – caso a pessoa tenha a sorte de pegar uma fila que anda mais rápido – porém há também um risco maior de esperar bastante, caso se tenha o azar de pegar uma fila mais demorada. Já no sistema de fila única a incerteza em relação à duração do tempo de espera é bem menor, pois muito provavelmente ele não se distanciará muito desse valor médio de 120 minutos. Isso ameniza em parte o aspecto negativo associado ao elevado tempo de espera, pois pelo menos a pessoa tem uma noção mais exata de sua duração.
Um segundo ponto que ilustro a seguir é a injustiça do sistema de filas separadas. Realizei simulações adicionais para esta configuração, observando o que poderia ocorrer com duas pessoas que estivessem em filas diferentes, na seguinte situação: a primeira na 40ª posição de uma das filas e a segunda em uma posição mais a frente (entre a 30ª até a 39ª) na fila ao lado. A idéia foi verificar o percentual de vezes em que a pessoa da primeira fila, mesmo estando mais atrás, ainda conseguiria ser atendida antes do que a pessoa da outra fila. Os resultados são mostrados na tabela abaixo:<!--[if gte mso 9]>     Normal   0         21         false   false   false      PT-BR   X-NONE   X-NONE                                                     MicrosoftInternetExplorer4                                                   <![endif]--><!--[if gte mso 9]>  <w:LatentStyles DefLockedState="false" DefUnhideWhenUsed="true"
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      Posição da pessoa   na segunda fila      39ª      37ª      35ª      32ª      30ª  
      Percentual das vezes em que uma   pessoa na 40ª posição da primeira fila ainda conseguiria ser atendida antes   da pessoa na segunda fila      46,8%      31,8%      23,5%      11,1%      3,9%  

Nota-se que em quase metade das vezes a pessoa da primeira fila seria atendida antes de uma pessoa 1 posição à frente na segunda fila, e em quase um quarto das vezes ela seria atendida antes da pessoa que estivesse 5 posições à frente na outra fila. Mesmo alguém na 30ª posição da outra fila ainda teria um risco próximo a 4% de ser atendido depois da pessoa da primeira fila, situada 10 posições atrás.
De qualquer forma, é importante ressaltar que o sistema de fila única não elimina o problema da pequena quantidade de caixas colocada para atender ao público do Rock In Rio, que nitidamente foi inferior ao que seria esperado para um evento destinado a receber 100 mil pessoas por dia._____________________________________________________________________________________ <!--[if gte mso 9]>     Normal   0         21         false   false   false      PT-BR   X-NONE   X-NONE                                                     MicrosoftInternetExplorer4                                                   <![endif]--><!--[if gte mso 9]>  <w:LatentStyles DefLockedState="false" DefUnhideWhenUsed="true"
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1) Nos exemplos citados de atendimento do Bob´s ou agências bancarias, é evidente que isto não ocorreria, pois se algum atendente ficasse ocioso as pessoas se deslocariam de uma fila para outra. Entretanto, em situações como um SAC por telefone onde houvesse vários números para se ligar diretamente, poderia haver fila em algum destes números, enquanto um outro número estivesse ocioso. Por este motivo, implantaram-se os atuais sistemas de atendimento onde há apenas um número, formando-se uma fila única imaginária (a famosa música que se ouve ao telefone) para distribuição dos clientes entre os vários ramais existentes. 
2) Em alguns modelos, como por exemplo quando se utiliza uma distribuição exponencial para representar tanto a chegada como a saída de clientes, é possível deduzir expressões teóricas exatas para as medidas de desempenho do sistema.
3) É evidente que não coletei minuciosamente dados reais para modelar este sistema. Apenas para ilustrar a análise, considerei que o tempo de atendimento seguia uma distribuição gama, com parâmetro r = 2 e valor médio de 3 minutos. Esta distribuição tem a função densidade de probabilidade mostrada abaixo, onde r e a são parâmetros positivos. O valor médio de X é dado pela razão r/a.