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Describe queuing system. What is the condition that leads to its stability?
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A queuing system is described by

• Calling population

• System capacity

• The arrival process

• Queue behaviour and Queue discipline

• Service time and service mechanism

The calling population:

• It is the population of potential customer those require service from system. It may be finite or infinite.

• System having large calling population is usually considered as infinite.

• Example: Customer at banks, restaurant. System having less and countable population is usually considered as finite.

• In infinite population model the arrival rate is not affected by the number of customer who have left the calling population and joined the queuing system.

• In an finite population model the arrival rate to the queuing system depend on number of customer being served and waiting.

System capacity:

• In many system there is a limit to the number of customer that may be in the waiting line or system.

• An arriving customer who finds the system full does not enters but returns to the calling population.

• Some system may be considered as having unlimited capacity example: ticket counter at station.

The arrival process:

• The arrival process for the model is characterised in terms of inter arrival times of successive customer.

• Arrival may occur at scheduled time or at random times. When arrival times are at random then inter arrival times are usually characterised by probability distribution and most important model for random arrivals is the Poisson process and for Poisson process arrival rate is ‘λ’ customers per unit time.

• In scheduled arrival inter arrival time of customers are constant.

Queue behaviour and Queue discipline:

• Queue discipline refers to the logical ordering of customers in a queue and determines which customer will be chosen for service when a server becomes free. OR

• Queue discipline refers to the rule that a server uses to choose the next customer from the queue when the server completes the service of the current customer.

• Common queue disciplines include first-in-first-out (FIFO); last-in-first-out (LIFO); service in random order (SIRO); shortest processing time first (SPT); and service according to priority (PR).

First in first out :

• This principle states that customers are served one at a time and that the customer that has been waiting the longest is served first.

Last in first out :

• This principle also serves customers one at a time, however the customer with the shortest waiting time will be served first. Also known as a stack.

Processor sharing:

• Service capacity is shared equally between customers.

Priority:

• Customers with high priority are served first.[17] Priority queues can be of two types,

• Non-pre emptive (where a job in service cannot be interrupted) and pre emptive (where a job in service can be interrupted by a higher priority job). No work is lost in either model.

Shortest job first:

• The next job to be served is the one with the smallest sizePre emptive shortest job first:

• The next job to be served is the one with the original smallest size.

Shortest remaining processing time:

• The next job to serve is the one with the smallest remaining processing requirement.

• Queue behaviour refer to customer action while in a queue waiting for service to begin.

Different queue behaviour are bulky, renege and jockey

Bulky: In this case customer can leave the queue if queue is too long.

Renege: In this case the customer can leave the queue if the queue is moving very slowly.

Jockey: In this case the customer can move from one queue to another.

Service time and service mechanism:

• Service mechanism can be described by service rate. Service may be single or in batches. Random services is characterised as a sequence of independent, identically distributed random variables.

• The exponential, weibull, gamma, log- normal and normal distributions are used as per situation.

• Service time may be identically distributed for all customers of same class different class customer may have different service time.

Stability condition:

• Denote by $N_0$(t) the number of available demands that have not been placed in the buffer. i.e. those that will return to the system.

• By the well-known terminology, these demands are on the orbit. Thus, there are $N_0$(t) + N(t) demands in the system.

• According to the definition by the stability of the given random process will be meant the uniform boundness in probability.

• Stability condition 1: For any given value of q such that q>0 the system is stable for any λ, r and a<∞ .Stability condition 2: if value of q=0, p=$λ_t$<1 then for any 0≤a≤∞ then we say that the system is stable.