Waiting Line Models Waiting takes place in virtually
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Waiting Line Models Waiting takes place in virtually every productive process or service. Since the time spent by people and things waiting in line is a valuable resource, the reduction of waiting time is an important aspect of operations management. Quality Service = Quick Service More service capacity = Less waiting time = Increased Cost Goal of Queuing Theory (another name for Waiting Line theory) is to find the trade-off point between the cost of improved service and the cost of making the customer wait.
Why do waiting lines form? n People or things arrive at the server faster than they can be served. n Customers arrive at random times, and the time required to serve each individually is not the same. n What are the cost relationships in Waiting Line Analysis? n As the level of service improves, the cost of service increases n Better service requires more servers
What are the elements of a waiting line? The Calling Population - the source of the customers to the queuing system: can be either infinite or finite. The Arrival Rate - the frequency at which customers arrive at a waiting line according to a probability distribution (average arrival rate is signified by l ) Service Time - the time required to serve a customer, is most frequently described by the negative exponential distribution ( average service rate is signified by m) Queue Discipline and Length - the order in which customers are served First come, first served Last in, first out Random Alphabetically
Basic Waiting Line Structures n Four basic structures of waiting lines, determined by the nature of the service facilities. Channels are the number of parallel servers, phases denote the number of sequential servers a customer must go through to receive service. n Single channel, single phase All customers go through single server one at a time for entire process n Single channel, multiple phase - All customer go through a series of servers one at a time to complete the process. n Multiple channel, single phase - All customers get split up into a group of servers one at a time for the entire process. n Multiple channel, multiple phase - All customers get split into a group servers and further proceed through a series of servers to complete the process.
Operating Characteristics n Notation Operating Characteristics . n L Average number of customers in the system n Lq Average number of customers in the waiting line n W Average time a customer spends in the system n Wq Average time a customer spends waiting in line n P 0 Probability of no (zero) customers in the system n Pn Probability of n customers in the system n r Utilization rate; proportion of time the system is in use.
Single-Channel, Single-Phase There are several variations of the single server waiting line system: n Poisson arrival rate, exponential service times n Poisson arrival rate, general (or unknown) distribution of service times n Poisson arrival rate, constant service times n Poisson arrival rate, exponential service times with a finite queue n Poisson arrival rate, exponential service times with a finite calling population
Single-Channel, Single-Phase In our single-server model, we will assume the following: n Poisson Arrival Rate Exponential Service n Times First-come, first-serve queue discipline n Infinite queue length n Infinite calling population
Single-Channel, Single-Phase The symbols which we will use are: n n l = mean arrival rate m = mean service rate
Single-Channel, Single-Phase FORMULAS n Probability that no customers are in queuing system, P 0 = ( 1 - l/m ) n Probability that exactly n customers in the system, Pn = ( l/m )n * P 0 n Average number of customers in the system, L = ( l / m-l ) n Average number of customers in the waiting line, Lq = ( l 2 / m(m-l) n Average time a customer spend in system, W = L / l n Average time customer spends waiting in line, Wq = ( l / m(m-l) ) n Probability that the server is busy (utilization factor), r = l/m n Probability that the server is idle, I = 1 - r
Multiple-Channel, Single-Phase In our multiple-server model, we will assume the following: n Poisson Arrival Rate n Exponential Service Times n First-come, first-serve queue discipline n Infinite queue length n Infinite calling population
Multiple-Channel, Single-Phase The symbols which we will us are: n l = mean arrival rate n m = mean service rate n s = number of servers
Multiple-Channel, Single-Phase FORMULAS n Probability that no customers are in queuing system, P 0 = look to Table n Probability that exactly n customers in the system, Pn = (1/s!sn-s)*( l/m )n * P 0 for n > s Pn = ( 1/n! )*( l/m )n * P 0 for n <= s n Average number of customers in the system, L = ( lm(l/m)s / (s-1)!(sm-l )2)P 0 + l/m n Average number of customers in the waiting line, Lq = L - l/m n Average time a customer spend in system, W = L / l n Average time customer spends waiting in line to be served, Wq = W - 1 / m n Probability that the server is busy (utilization factor), r = l/sm n Probability that the server is idle and customer can be served, I = 1 - r
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