^{1}

^{1}

^{2}

^{3}

^{4}

^{1}

^{2}

^{3}

^{4}

In many real-life queueing systems, the servers are often heterogeneous, namely they work at different rates. This paper provides a simple method to compute tight upper bounds on two important performance measures of single-class heterogeneous multi-server Markovian queueing systems, namely the average number in queue and the average waiting time in queue. This method is based on an expansion of the state space that is followed by an approximate reduction of the state space, only considering the most probable states. In most cases tested, we were able to approximate the actual behavior of the system with smaller errors than those obtained from traditional homogeneous multiserver Markovian queues, as shown by GPSS simulations. In addition, we have correlated the quality of the approximation with the degree of heterogeneity of the system, which was evaluated using its Gini index. Finally, we have shown that the bounds are robust and still useful, even considering quite different allocation strategies. A large number of simulation results show the accuracy of the proposed method that is better than that of classical homogeneous multiserver Markovian formulae in many situations.

A better understanding of queueing systems is of paramount importance in order to improve their applications, which is the scope of the current study. Markov and semi-Markov processes are among the stochastic-based methods traditionally used for performance evaluation of multistate systems [

A single-class heterogeneous multiserver queue.

The aims of this paper are twofold. First, we will derive upper bounds on performance measures of single-class heterogeneous multi-server Markovian queueing systems see Figure

The paper is organized as follows. In Section

Queues are ordinary phenomena that happen all the time. They can be encountered everywhere. Everyone has already joined a queue at least once, for instance, when driving home and lining up in a traffic jam, when paying bills or when buying a snack. Day-to-day queues are very frequent, and they are not even perceived in some cases. Queues happen, for example, throughout manufacturing processes [

As mentioned earlier, the focus here is on heterogeneous multi-server single queues. The importance of modeling such systems comes from their similarities with real-life systems. Indeed, many real situations involve servers working at different rates. To illustrate such a situation, let us consider manual assembly, in which human beings can be seen as servers. As noticed by Wang et al. [

In the literature, there exist several studies that have taken into account the heterogeneity of the servers. One of the first studies that were developed for queueing systems considering the differences in the processing capacity of the servers was done by Gumbel [

Singh [

Gall [

Grassmann and Zhao [

Boxma et al. [

Chao and Luh [

Marmony [

The multiclass multi-server (MCMS) system is a more complex model of a queueing system. The MCMS system presents different types of jobs arriving in a system with multiple servers. If we consider the servers to be heterogeneous, the MCMS system constitutes a generalization of the queueing system of interest in this paper, which only assumes one type of job. Van Harten and Sleptchenko [

Finally, it is also worthwhile mentioning that there are a number of papers in the literature focusing on controlling heterogeneous server queues using different allocation strategies in order to optimize their performance measures as shown in the work of Lin and Kumar [

The formulation proposed in this paper is fundamentally based on the equilibrium equations of the system that are obtained from the conservation of flow and some approximations. For our convenience and without loss of generality, the indices

Considering heavy traffic conditions, it is intuitive that if there is a job in the system, it will most likely be on the slower server. Thus, assuming heavy traffic conditions the probability to find a job in the system in the slowest server is higher than the probability to find it in any other server, because in average, the job will stay longer in a slow server than in a fast one. Another argument that supports the statement that a job in the system is highly probable to be found in the slowest server is the fact that the exponential distribution has no memory. Also, in order to know which server is more likely to be the last one to finish the work at any time

State transition diagram for

Figure

Indeed, under the assumptions of (

Thus, (

When

For a better understanding of the influence of the

In (

Developing the equilibrium equations, it can be shown that the probability

In order to compute the probability

In order to keep the system stable (in other words, in order to prevent the queue from growing indefinitely), the system utilization

As

Substituting (

It is possible to obtain a formulation of the performance measures of the system from the probabilities

Substituting (

Using the change of indices

The following equation for the average waiting time in queue,

Although it is recognized that in some cases discrete event simulation techniques are less suitable because of the high computational capacities required, such techniques play a key role in queueing systems analysis [

It is important to note that many simulation model limitations are encountered in our work. We experienced some difficulties in obtaining accurate results without too many computational efforts and during the generalization of the results. In addition, the required high accuracy led to some technical difficulties. The required level of accuracy dramatically increases the simulation running times as well as the number of replications (large

Finally, it is important to note that we have encountered some problems in the generalization of the results. Each model simulated gave results that were only valid for the specific system and combination of parameters to which they were related. Therefore, it was necessary to simulate many different configurations with slightly different parameters in order to get a complete understanding with the required accuracy of the general behavior of the

In this section, we present our experimental results in order to validate the quality of the proposed approximation and demonstrate that the proposed upper bounds can be used to get an estimate of the performance measures of

To do so, it is necessary to thoroughly understand the proposed model. Thus, several types of queueing systems were created by changing for each one of them at least one of the following parameters:

the number of servers

the arrival rate of jobs,

the level of heterogeneity of the servers, given by the corresponding Gini Index.

The level of heterogeneity of the servers indicates how the overall processing capacity of the systems has been distributed among the servers. In this paper, we have chosen the Gini index of inequality to measure the differences between the processing capacities of the servers for a given system (see, for instance, Shalit [

The average waiting times in queue

Average waiting times in queue for a heavily loaded system,

Average waiting times in queue for a moderately loaded system,

Using the FSF allocation strategy, the systems can show a region for which the average waiting time in queue is optimum, namely,

In order to better compare our approximation with the traditional homogeneous

Figures

Maximum errors (in %) in the predictions of the average waiting times in queue for the proposed method and for the traditional homogeneous

Number of servers | Allocation strategy | ||||

Approximation method | FSF | RCS | SSF | ||

2 | 0.90 | Proposed bounds | 1.36 | 0.77 | |

Homogeneous | 4.66 | 4.76 | |||

0.75 | Proposed bounds | 6.91 | 3.93 | 1.93 | |

Homogeneous | 12.52 | 13.11 | 13.49 | ||

0.60 | Proposed bounds | 7.31 | 3.18 | ||

Homogeneous | 21.49 | 22.95 | |||

3 | 0.90 | Proposed bounds | 4.30 | 2.52 | 1.37 |

Homogeneous | 8.42 | 8.88 | 8.87 | ||

0.75 | Proposed bounds | 13.70 | 7.40 | 3.51 | |

Homogeneous | 22.62 | 23.33 | 23.59 | ||

0.60 | Proposed bounds | 29.56 | 14.64 | 5.88 | |

Homogeneous | 37.97 | 39.21 | 39.79 | ||

6 | 0.90 | Proposed bounds | 6.89 | 4.70 | 2.45 |

Homogeneous | 16.37 | 16.57 | 17.16 | ||

0.75 | Proposed bounds | 23.58 | 11.82 | 5.43 | |

Homogeneous | 40.41 | 41.81 | 42.36 | ||

0.60 | Proposed bounds | 62.75 | 22.64 | 8.64 | |

Homogeneous | 57.97 | 64.93 | 65.66 | ||

12 | 0.90 | Proposed bounds | 8.63 | 4.35 | 2.22 |

Homogeneous | 27.48 | 27.96 | 28.28 | ||

0.75 | Proposed bounds | 31.77 | 13.69 | 5.85 | |

Homogeneous | 62.09 | 63.17 | 63.53 | ||

0.60 | Proposed bounds | 99.33 | 27.06 | 9.46 | |

Homogeneous | 85.53 | 86.54 | 86.87 |

Errors in the estimation of the average waiting times in queue for the traditional homogeneous

Errors in the estimation of the average waiting times in queue for the traditional homogeneous

In Figures

In other words, this seems to indicate that the proposed bounds give more rapidly better predictions than the traditional homogeneous

In Table

In general, the errors increase when the number of servers increases. In Table

Finally, Table

Average errors (in %) in the predictions of the average waiting times in queue for the proposed method and for the traditional homogeneous

Number of servers | Allocation strategy | ||||

Approximation method | FSF | RCS | SSF | ||

2 | 0.90 | Proposed bounds | 1.31 | 0.82 | 0.46 |

Homogeneous | 1.03 | 1.45 | 1.80 | ||

0.75 | Proposed bounds | 4.02 | 2.33 | 1.16 | |

Homogeneous | 2.75 | 3.93 | 5.01 | ||

0.60 | Proposed bounds | 8.10 | 4.28 | 1.91 | |

Homogeneous | 4.58 | 6.91 | 8.99 | ||

3 | 0.90 | Proposed bounds | 2.54 | 1.53 | 0.87 |

Homogeneous | 2.15 | 2.96 | 3.59 | ||

0.75 | Proposed bounds | 7.89 | 4.43 | 2.11 | |

Homogeneous | 5.62 | 8.07 | 10.09 | ||

0.60 | Proposed bounds | 16.75 | 8.60 | 3.49 | |

Homogeneous | 9.38 | 14.00 | 17.98 | ||

6 | 0.90 | Proposed bounds | 3.15 | 2.12 | 0.88 |

Homogeneous | 5.35 | 6.20 | 7.18 | ||

0.75 | Proposed bounds | 10.74 | 5.17 | 2.49 | |

Homogeneous | 13.68 | 17.21 | 19.24 | ||

0.60 | Proposed bounds | 26.19 | 10.27 | 4.22 | |

Homogeneous | 21.72 | 29.39 | 33.01 | ||

12 | 0.90 | Proposed bounds | 4.01 | 1.75 | 0.84 |

Homogeneous | 13.43 | 15.27 | 16.00 | ||

0.75 | Proposed bounds | 15.51 | 5.75 | 2.40 | |

Homogeneous | 33.26 | 38.56 | 40.32 | ||

0.60 | Proposed bounds | 49.91 | 11.83 | 4.33 | |

Homogeneous | 48.30 | 60.44 | 62.51 |

In many practical situations, queueing theory has been successfully applied [

Future possible research in this field involves the development of tight lower bounds for the performance measures and extensions to general arrivals, batch arrivals, general service times, and finite queues. Also, it is important to consider that the lifetime of each server is finite in real life. The investigation of the effect of finite lifetimes in the performance of the server allocation strategies is another interesting topic for future research in the area.

This research has been partially funded by CNPq (Conselho Nacional de Desenvolvimento Científico e Tecnológico; Grants nos. 201046/1994-6, 301809/1996-8, 307702/2004-9, 472066/2004-8, 304944/2007-6, 561259/2008-9, 553019/2009-0, 550207/2010-4, 501532/2010-2, 303388/2010-2), by CAPES (Coordenação de Aperfeiçoamento de Pessoal de Nível Superior; Grant no BEX-0522/07-4), and by FAPEMIG (Grants no CEX-289/98, CEX-855/98, TEC-875/07, CEX-PPM-00401/08, and CEX-PPM-00390-10).